Notation by work
Streaming Polyphase Factor Decomposition / SKAT
All notation · Glossary · Source coverage
319 symbol spellings in 481 mathematical passages across 6 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 319 entries are available below.
+Introduction passage · 30 passages
Search aliases: +
Introduction passage · FINAL_MANUSCRIPT.tex, line 1412
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\max\{0,\widetilde\sigma_i(B)-\delta\}^2 \le \lambda_i(AA^\top) \le \left( \widetilde\sigma_i(B)+\delta \right)^2.
=Introduction passage · 133 passages
Search aliases: =
Introduction passage · FINAL_MANUSCRIPT.tex, line 740
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
e=\{h,g\}
AIntroduction passage · 54 passages
Search aliases: A
Introduction passage · FINAL_MANUSCRIPT.tex, line 1397
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
A
A^\topIntroduction passage · 21 passages
Search aliases: A^top top
Introduction passage · FINAL_MANUSCRIPT.tex, line 1412
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\max\{0,\widetilde\sigma_i(B)-\delta\}^2 \le \lambda_i(AA^\top) \le \left( \widetilde\sigma_i(B)+\delta \right)^2.
BIntroduction passage · 30 passages
Search aliases: B
Introduction passage · FINAL_MANUSCRIPT.tex, line 1398
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\widetilde\sigma_i(B)
\mathcal BIntroduction passage · 35 passages
Search aliases: B mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 739
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal B
\mathcal B_{\mathcal O}Introduction passage · 4 passages
Search aliases: B_ O mathcal mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 747
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal B_{\mathcal O} =\mathcal B-\sum_{e\in\mathcal O}\Delta_e.
\mathcal B_{gh}Introduction passage · 1 passage
Search aliases: B_gh mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 741
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
(\Delta_e)_{hg}=\mathcal B_{hg},\qquad (\Delta_e)_{gh}=\mathcal B_{gh},
B_hIntroduction passage · 1 passage
Search aliases: B_h
Introduction passage · FINAL_MANUSCRIPT.tex, line 2215
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
\mathcal B_{hg}Introduction passage · 6 passages
Search aliases: B_hg mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 741
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
(\Delta_e)_{hg}=\mathcal B_{hg},\qquad (\Delta_e)_{gh}=\mathcal B_{gh},
cIntroduction passage · 25 passages
Search aliases: c
Introduction passage · FINAL_MANUSCRIPT.tex, line 931
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
c_e\ge0
c_eIntroduction passage · 2 passages
Search aliases: c_e
Introduction passage · FINAL_MANUSCRIPT.tex, line 931
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
c_e\ge0
\mathcal EIntroduction passage · 5 passages
Search aliases: E mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 738
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal E
eIntroduction passage · 28 passages
Search aliases: e
Introduction passage · FINAL_MANUSCRIPT.tex, line 740
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
e=\{h,g\}
gIntroduction passage · 28 passages
Search aliases: g
Introduction passage · FINAL_MANUSCRIPT.tex, line 740
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
e=\{h,g\}
HIntroduction passage · 70 passages
Search aliases: H
Introduction passage · FINAL_MANUSCRIPT.tex, line 739
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
H
hIntroduction passage · 40 passages
Search aliases: h
Introduction passage · FINAL_MANUSCRIPT.tex, line 740
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
e=\{h,g\}
iIntroduction passage · 22 passages
Search aliases: i
Introduction passage · FINAL_MANUSCRIPT.tex, line 1397
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\sigma_i(A)
JIntroduction passage · 4 passages
Search aliases: J
Introduction passage · FINAL_MANUSCRIPT.tex, line 2215
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
J_{h1}Introduction passage · 1 passage
Search aliases: J_h1
Introduction passage · FINAL_MANUSCRIPT.tex, line 2215
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
J_{hB_h}Introduction passage · 1 passage
Search aliases: J_hB_h
Introduction passage · FINAL_MANUSCRIPT.tex, line 2215
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
kIntroduction passage · 52 passages
Search aliases: k
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
mIntroduction passage · 20 passages
Search aliases: m
Introduction passage · FINAL_MANUSCRIPT.tex, line 2211
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\maxIntroduction passage · 8 passages
Search aliases: max
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\max_{x_e\in\{0,1\}}Introduction passage · 1 passage
Search aliases: max_x_e∈{0,1} max in
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\mathcal OIntroduction passage · 14 passages
Search aliases: O mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 745
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal O\subseteq\mathcal E
PIntroduction passage · 28 passages
Search aliases: P
Introduction passage · FINAL_MANUSCRIPT.tex, line 2221
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
P_h
\mathcal PIntroduction passage · 2 passages
Search aliases: P mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 2211
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\mathcal P_1Introduction passage · 1 passage
Search aliases: P_1 mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 2211
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
P_gIntroduction passage · 17 passages
Search aliases: P_g
Introduction passage · FINAL_MANUSCRIPT.tex, line 2246
Storage refinement changes only the order and grouping of finite sums. By partition refinement invariance, the streamed products reconstruct the same A, A^top, and therefore the same H=A^top A. Since each P_h is fixed by the structural phase partition, every block P_hHP_g is unchanged. The phase graph is defined only by these blocks, so its path structure and light cones are unchanged. Finally, an exact matrix-free Krylov method depends only on repeated applications of the same operator to the s…
Exact mathematical passage
P_hHP_g
P_hIntroduction passage · 23 passages
Search aliases: P_h
Introduction passage · FINAL_MANUSCRIPT.tex, line 2221
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
P_h
\mathcal P_HIntroduction passage · 1 passage
Search aliases: P_H mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 2211
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\mathcal P_hIntroduction passage · 1 passage
Search aliases: P_h mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 2215
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
SIntroduction passage · 14 passages
Search aliases: S
Introduction passage · FINAL_MANUSCRIPT.tex, line 2220
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
S
\sqcupIntroduction passage · 3 passages
Search aliases: sqcup
Introduction passage · FINAL_MANUSCRIPT.tex, line 2211
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\topIntroduction passage · 56 passages
Search aliases: top
Introduction passage · FINAL_MANUSCRIPT.tex, line 1412
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\max\{0,\widetilde\sigma_i(B)-\delta\}^2 \le \lambda_i(AA^\top) \le \left( \widetilde\sigma_i(B)+\delta \right)^2.
\widetilde\sigmaIntroduction passage · 3 passages
Search aliases: widetildeσ widetilde sigma
Introduction passage · FINAL_MANUSCRIPT.tex, line 1398
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\widetilde\sigma_i(B)
\widetilde\sigma_iIntroduction passage · 3 passages
Search aliases: widetildeσ_i widetilde sigma
Introduction passage · FINAL_MANUSCRIPT.tex, line 1398
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\widetilde\sigma_i(B)
xIntroduction passage · 10 passages
Search aliases: x
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
x_eIntroduction passage · 1 passage
Search aliases: x_e
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
|Introduction passage · 4 passages
Search aliases: |
Introduction passage · FINAL_MANUSCRIPT.tex, line 1400
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\boxed{ \left| \sigma_i(A) - \widetilde\sigma_i(B) \right| \le \delta. }
\betaIntroduction passage · 12 passages
Search aliases: β beta
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\beta_e^{(k)}Introduction passage · 6 passages
Search aliases: β_e^(k) beta
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\DeltaIntroduction passage · 6 passages
Search aliases: Δ Delta
Introduction passage · FINAL_MANUSCRIPT.tex, line 740
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\Delta_e
\deltaIntroduction passage · 4 passages
Search aliases: δ delta
Introduction passage · FINAL_MANUSCRIPT.tex, line 1400
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\boxed{ \left| \sigma_i(A) - \widetilde\sigma_i(B) \right| \le \delta. }
\Delta_eIntroduction passage · 6 passages
Search aliases: Δ_e Delta
Introduction passage · FINAL_MANUSCRIPT.tex, line 740
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\Delta_e
\lambdaIntroduction passage · 10 passages
Search aliases: λ lambda
Introduction passage · FINAL_MANUSCRIPT.tex, line 1412
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\max\{0,\widetilde\sigma_i(B)-\delta\}^2 \le \lambda_i(AA^\top) \le \left( \widetilde\sigma_i(B)+\delta \right)^2.
\lambda_iIntroduction passage · 6 passages
Search aliases: λ_i lambda
Introduction passage · FINAL_MANUSCRIPT.tex, line 1412
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\max\{0,\widetilde\sigma_i(B)-\delta\}^2 \le \lambda_i(AA^\top) \le \left( \widetilde\sigma_i(B)+\delta \right)^2.
\sigmaIntroduction passage · 2 passages
Search aliases: σ sigma
Introduction passage · FINAL_MANUSCRIPT.tex, line 1397
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\sigma_i(A)
\sigma_iIntroduction passage · 2 passages
Search aliases: σ_i sigma
Introduction passage · FINAL_MANUSCRIPT.tex, line 1397
[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\sigma_i(A)
\tauIntroduction passage · 10 passages
Search aliases: τ tau
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\ldotsIntroduction passage · 6 passages
Search aliases: … ldots
Introduction passage · FINAL_MANUSCRIPT.tex, line 2211
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\inIntroduction passage · 21 passages
Search aliases: ∈ in
Introduction passage · FINAL_MANUSCRIPT.tex, line 747
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal B_{\mathcal O} =\mathcal B-\sum_{e\in\mathcal O}\Delta_e.
\sumIntroduction passage · 38 passages
Search aliases: ∑ sum
Introduction passage · FINAL_MANUSCRIPT.tex, line 747
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal B_{\mathcal O} =\mathcal B-\sum_{e\in\mathcal O}\Delta_e.
\sum_eIntroduction passage · 1 passage
Search aliases: ∑_e sum
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\sum_{e\in\mathcal O}Introduction passage · 7 passages
Search aliases: ∑_e∈ O sum in mathcal
Introduction passage · FINAL_MANUSCRIPT.tex, line 747
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal B_{\mathcal O} =\mathcal B-\sum_{e\in\mathcal O}\Delta_e.
\leIntroduction passage · 35 passages
Search aliases: ≤ le
Introduction passage · FINAL_MANUSCRIPT.tex, line 934
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\geIntroduction passage · 9 passages
Search aliases: ≥ ge
Introduction passage · FINAL_MANUSCRIPT.tex, line 931
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic; its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
c_e\ge0
\subseteqIntroduction passage · 3 passages
Search aliases: ⊆ subseteq
Introduction passage · FINAL_MANUSCRIPT.tex, line 745
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.
Exact mathematical passage
\mathcal O\subseteq\mathcal E
\cdotsIntroduction passage · 6 passages
Search aliases: ⋯ cdots
Introduction passage · FINAL_MANUSCRIPT.tex, line 2211
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
aUse in source · 7 passages
Search aliases: a
Use in source · FINAL_MANUSCRIPT.tex, line 647
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
A_1Use in source · 2 passages
Search aliases: A_1
Use in source · FINAL_MANUSCRIPT.tex, line 163
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
A=[A_1\ \cdots\ A_B].
A_1^\topUse in source · 1 passage
Search aliases: A_1^top top
Use in source · FINAL_MANUSCRIPT.tex, line 187
Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
[A_1\ \cdots\ A_B] \begin{bmatrix} A_1^\top\\ \vdots\\ A_B^\top \end{bmatrix} = \sum_bA_bA_b^\top.
A_BUse in source · 2 passages
Search aliases: A_B
Use in source · FINAL_MANUSCRIPT.tex, line 163
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
A=[A_1\ \cdots\ A_B].
A_bUse in source · 6 passages
Search aliases: A_b
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
A_{b,I_r}Use in source · 2 passages
Search aliases: A_b,I_r
Use in source · FINAL_MANUSCRIPT.tex, line 363
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ A_{b,I_r} = H_{b,I_r}-Q_{X,I_r}R_b. }
A_{b,I_s}^\topUse in source · 1 passage
Search aliases: A_b,I_s^top top
Use in source · FINAL_MANUSCRIPT.tex, line 371
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ K_{I_r,I_s} = \sum_b A_{b,I_r}A_{b,I_s}^\top. }
A_B^\topUse in source · 1 passage
Search aliases: A_B^top top
Use in source · FINAL_MANUSCRIPT.tex, line 187
Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
[A_1\ \cdots\ A_B] \begin{bmatrix} A_1^\top\\ \vdots\\ A_B^\top \end{bmatrix} = \sum_bA_bA_b^\top.
A_b^\topUse in source · 4 passages
Search aliases: A_b^top top
Use in source · FINAL_MANUSCRIPT.tex, line 168
[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.
Exact mathematical passage
\boxed{ K = AA^\top = \sum_{b=1}^B A_bA_b^\top = M_X \left( \sum_{b=1}^B G_bW_bG_b^\top \right) M_X. }
a_dUse in source · 2 passages
Search aliases: a_d
Use in source · FINAL_MANUSCRIPT.tex, line 828
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
A_JUse in source · 1 passage
Search aliases: A_J
Use in source · FINAL_MANUSCRIPT.tex, line 399
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
a_jUse in source · 2 passages
Search aliases: a_j
Use in source · FINAL_MANUSCRIPT.tex, line 647
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
A_{J'}Use in source · 1 passage
Search aliases: A_J'
Use in source · FINAL_MANUSCRIPT.tex, line 399
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
A_{J'}^\topUse in source · 1 passage
Search aliases: A_J'^top top
Use in source · FINAL_MANUSCRIPT.tex, line 399
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
A_J^\topUse in source · 1 passage
Search aliases: A_J^top top
Use in source · FINAL_MANUSCRIPT.tex, line 399
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
bUse in source · 32 passages
Search aliases: b
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
\mathcal B^jUse in source · 4 passages
Search aliases: B^j mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 651
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
\boxed{ \rho\bigl((p(H)-p(H_0))X\bigr) \leq \sum_{j=1}^d |a_j|(\mathcal B^j-\mathcal B_0^j)\rho(X). }
\mathcal B^kUse in source · 16 passages
Search aliases: B^k mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 522
[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).
Exact mathematical passage
\boxed{ \rho(H^kX)\leq \mathcal B^k\rho(X). }
\mathcal B^{k-1-j}Use in source · 4 passages
Search aliases: B^k-1-j mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 588
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
\rho\bigl((H^k-H_0^k)X\bigr) \leq \sum_{j=0}^{k-1} \mathcal B^{k-1-j}\mathcal R\mathcal B_0^j\rho(X).
B^\topUse in source · 3 passages
Search aliases: B^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1159
[Exact range core] The nonzero eigenvalues of K=AA^top are exactly the eigenvalues of BB^top.
Exact mathematical passage
BB^\top.
\mathcal B_{\mathcal O}^jUse in source · 2 passages
Search aliases: B_ O^j mathcal mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 801
The first inequality in the second display is Theorem~[source reference]. For the transfer matrices, the exact noncommutative telescoping identity is B^k- B_ O^k = ∑_j=0^k-1 B^k-1-j ( B- B_ O) B_ O^j. All matrices and ρ_0 are nonnegative, and 0≤ B_ O≤ B entrywise. Hence B_ O^jρ_0 ≤ B^jρ_0. Substituting B- B_ O = ∑_e∈ OΔ_e and distributing the sum gives the componentwise majorant. The final scalar bound follows from the Euclidean triangle inequality.
Exact mathematical passage
\mathcal B^k-\mathcal B_{\mathcal O}^k = \sum_{j=0}^{k-1} \mathcal B^{k-1-j} (\mathcal B-\mathcal B_{\mathcal O}) \mathcal B_{\mathcal O}^j.
\mathcal B_{\mathcal O}^kUse in source · 5 passages
Search aliases: B_ O^k mathcal mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 775
[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).
Exact mathematical passage
\boxed{ 0\le (\mathcal B^k-\mathcal B_{\mathcal O}^k)\rho_0 \le \sum_{e\in\mathcal O}q_e^{(k)} }
\mathcal B_0Use in source · 3 passages
Search aliases: B_0 mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 547
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.
Exact mathematical passage
(\mathcal B_0)_{hg}= \begin{cases} \mathcal B_{hg},&(h,g)\in\mathcal E_0,\\ 0,&\text{otherwise}, \end{cases} \qquad \mathcal R=\mathcal B-\mathcal B_0.
\mathcal B_0^jUse in source · 3 passages
Search aliases: B_0^j mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 588
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
\rho\bigl((H^k-H_0^k)X\bigr) \leq \sum_{j=0}^{k-1} \mathcal B^{k-1-j}\mathcal R\mathcal B_0^j\rho(X).
\mathcal B_0^kUse in source · 8 passages
Search aliases: B_0^k mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 562
[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.
Exact mathematical passage
\boxed{ \rho\bigl((H^k-H_0^k)X\bigr) \leq (\mathcal B^k-\mathcal B_0^k)\rho(X). }
\bar GUse in source · 1 passage
Search aliases: bar G bar
Use in source · FINAL_MANUSCRIPT.tex, line 1888
G_b → G_b-bar G_b → variance scaling → M_X G_b → W_b^1/2 → A_bA_b^top,
Exact mathematical passage
G_b \to G_b-\bar G_b \to \text{variance scaling} \to M_X G_b \to W_b^{1/2} \to A_bA_b^\top,
\bar G_bUse in source · 1 passage
Search aliases: bar G_b bar
Use in source · FINAL_MANUSCRIPT.tex, line 1888
G_b → G_b-bar G_b → variance scaling → M_X G_b → W_b^1/2 → A_bA_b^top,
Exact mathematical passage
G_b \to G_b-\bar G_b \to \text{variance scaling} \to M_X G_b \to W_b^{1/2} \to A_bA_b^\top,
CUse in source · 10 passages
Search aliases: C
Use in source · FINAL_MANUSCRIPT.tex, line 1107
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp.
\mathcal CUse in source · 7 passages
Search aliases: C mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 1199
Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.
Exact mathematical passage
\mathcal C_{hg} = \|P_hEP_g\|_2.
\mathcal C_{\mathcal O}Use in source · 1 passage
Search aliases: C_ O mathcal mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 1338
A production schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.
Exact mathematical passage
\boxed{ \sum_{e\in\mathcal O}\beta_e^{(k)} \le \tau_K, \qquad \|\mathcal C_{\mathcal O}\|_2 \le \tau_\lambda. }
c_BUse in source · 1 passage
Search aliases: c_B
Use in source · FINAL_MANUSCRIPT.tex, line 1005
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
c_bUse in source · 4 passages
Search aliases: c_b
Use in source · FINAL_MANUSCRIPT.tex, line 996
The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.
Exact mathematical passage
c_b
\mathcal C_{hg}Use in source · 1 passage
Search aliases: C_hg mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 1199
Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.
Exact mathematical passage
\mathcal C_{hg} = \|P_hEP_g\|_2.
\mathcal C_{\rm complete}Use in source · 1 passage
Search aliases: C_rm complete mathcal rm
Use in source · FINAL_MANUSCRIPT.tex, line 1776
D . Consequently, the nonzero singular values of A are exactly the singular values of the stacked completion core C_rm complete = B D .
Exact mathematical passage
\mathcal C_{\rm complete} = \begin{bmatrix} B\\D \end{bmatrix}.
DUse in source · 5 passages
Search aliases: D
Use in source · FINAL_MANUSCRIPT.tex, line 828
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
\mathcal DUse in source · 1 passage
Search aliases: D mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 2191
Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structural propagation, the third controls spectral dimension, and the fourth controls deep-tail transform cost.
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
dUse in source · 9 passages
Search aliases: d
Use in source · FINAL_MANUSCRIPT.tex, line 647
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
\mathcal D_{\rm phase}Use in source · 1 passage
Search aliases: D_rm phase mathcal rm
Use in source · FINAL_MANUSCRIPT.tex, line 2191
Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structural propagation, the third controls spectral dimension, and the fourth controls deep-tail transform cost.
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\mathcal D_{\rm rank}Use in source · 1 passage
Search aliases: D_rm rank mathcal rm
Use in source · FINAL_MANUSCRIPT.tex, line 2191
Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structural propagation, the third controls spectral dimension, and the fourth controls deep-tail transform cost.
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\mathcal D_{\rm spectrum}Use in source · 1 passage
Search aliases: D_rm spectrum mathcal rm
Use in source · FINAL_MANUSCRIPT.tex, line 2191
Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structural propagation, the third controls spectral dimension, and the fourth controls deep-tail transform cost.
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\mathcal D_{\rm storage}Use in source · 1 passage
Search aliases: D_rm storage mathcal rm
Use in source · FINAL_MANUSCRIPT.tex, line 2191
Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structural propagation, the third controls spectral dimension, and the fourth controls deep-tail transform cost.
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\detUse in source · 1 passage
Search aliases: det
Use in source · FINAL_MANUSCRIPT.tex, line 1555
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
K_H(\theta) =-\frac12\log\det(I-2\theta H)
\operatorname{diag}Use in source · 1 passage
Search aliases: diag operatorname
Use in source · FINAL_MANUSCRIPT.tex, line 100
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
\downarrowUse in source · 1 passage
Search aliases: downarrow
Use in source · FINAL_MANUSCRIPT.tex, line 1805
The resulting production logic is not ``approximate forever.'' It is a certified handoff between two regimes: c streamed / pruned Krylov [2mm] downarrow [-1mm] residual moments (t,v,ρ) [2mm] downarrow [-1mm] continue Krylov, · residual range still large, exact streamed residual completion, · completion rank affordable.
Exact mathematical passage
\boxed{ \begin{array}{c} \text{streamed / pruned Krylov}\\[2mm] \downarrow\\[-1mm] \text{residual moments }(t,v,\rho)\\[2mm] \downarrow\\[-1mm] \begin{cases} \text{continue Krylov}, & \text{residual range still large},\\ \text{exact streamed residual completion}, & \text{completion rank affordable}. \end{cases} \end{array} }
EUse in source · 8 passages
Search aliases: E
Use in source · FINAL_MANUSCRIPT.tex, line 580
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
E=H-H_0
e^\topUse in source · 1 passage
Search aliases: e^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1935
item directly through the factor, Q=frac12 A^top e _2^2; item through the sample kernel, Q=frac12 e^top Ke; item through the eigencoordinates of K.
Exact mathematical passage
Q=\frac12 e^\top Ke;
\mathcal E_0Use in source · 2 passages
Search aliases: E_0 mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 543
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,
Exact mathematical passage
H_0=\sum_{(h,g)\in\mathcal E_0}P_hHP_g.
\expUse in source · 1 passage
Search aliases: exp
Use in source · FINAL_MANUSCRIPT.tex, line 1617
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat K_H(\theta)-\theta q\}, }
FUse in source · 29 passages
Search aliases: F
Use in source · FINAL_MANUSCRIPT.tex, line 503
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
fUse in source · 3 passages
Search aliases: f
Use in source · FINAL_MANUSCRIPT.tex, line 1030
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm fixed}
F^2Use in source · 1 passage
Search aliases: F^2
Use in source · FINAL_MANUSCRIPT.tex, line 1525
This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the residual spectrum is; the third is its largest possible coefficient.
Exact mathematical passage
\boxed{ t=\|F\|_F^2, \qquad v=\|FF^\top\|_F^2, \qquad \rho=\|F\|_2^2. }
F^\topUse in source · 3 passages
Search aliases: F^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1495
[Exact core--residual decomposition] A^top A = B^top B+F^top F. In particular the unresolved spectral term R=F^top F is positive semidefinite.
Exact mathematical passage
\boxed{ A^\top A = B^\top B+F^\top F. }
F_0Use in source · 3 passages
Search aliases: F_0
Use in source · FINAL_MANUSCRIPT.tex, line 2269
[Exact frontier-to-I/O reduction] Suppose the exact phase graph of H is used and the current Krylov block is supported on a phase set F_0. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vector. Such a block need not be loaded during that step.
Exact mathematical passage
F_0
F_kUse in source · 3 passages
Search aliases: F_k
Use in source · FINAL_MANUSCRIPT.tex, line 2269
[Exact frontier-to-I/O reduction] Suppose the exact phase graph of H is used and the current Krylov block is supported on a phase set F_0. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vector. Such a block need not be loaded during that step.
Exact mathematical passage
F_k
GUse in source · 20 passages
Search aliases: G
Use in source · FINAL_MANUSCRIPT.tex, line 61
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
A=M_XGW^{1/2}, \qquad K=AA^\top,
G_{[:,J_b]}Use in source · 1 passage
Search aliases: G_[:,J_b]
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
G_bUse in source · 9 passages
Search aliases: G_b
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
G_b^\topUse in source · 6 passages
Search aliases: G_b^top top
Use in source · FINAL_MANUSCRIPT.tex, line 168
[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.
Exact mathematical passage
\boxed{ K = AA^\top = \sum_{b=1}^B A_bA_b^\top = M_X \left( \sum_{b=1}^B G_bW_bG_b^\top \right) M_X. }
H^kUse in source · 14 passages
Search aliases: H^k
Use in source · FINAL_MANUSCRIPT.tex, line 449
[Polyphase light cone] If the graph distance from phase g to phase h exceeds k, then P_hH^kP_g=0.
Exact mathematical passage
\boxed{ P_hH^kP_g=0. }
H^{k-1-j}Use in source · 1 passage
Search aliases: H^k-1-j
Use in source · FINAL_MANUSCRIPT.tex, line 581
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
H^k-H_0^k = \sum_{j=0}^{k-1}H^{k-1-j}EH_0^j.
H_{\mathcal O}Use in source · 1 passage
Search aliases: H_ O mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 840
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
\boxed{ \lVert(p(H)-p(H_{\mathcal O}))X\rVert_F \le \sum_{e\in\mathcal O}\beta_e^{(p)}. }
H_{\mathcal O}^kUse in source · 1 passage
Search aliases: H_ O^k mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 784
[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).
Exact mathematical passage
\boxed{ \lVert(H^k-H_{\mathcal O}^k)X\rVert_F \le \left\lVert (\mathcal B^k-\mathcal B_{\mathcal O}^k)\rho_0 \right\rVert_2 \le \sum_{e\in\mathcal O}\beta_e^{(k)}. }
H_0Use in source · 12 passages
Search aliases: H_0
Use in source · FINAL_MANUSCRIPT.tex, line 543
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,
Exact mathematical passage
H_0=\sum_{(h,g)\in\mathcal E_0}P_hHP_g.
H_0^jUse in source · 1 passage
Search aliases: H_0^j
Use in source · FINAL_MANUSCRIPT.tex, line 581
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
H^k-H_0^k = \sum_{j=0}^{k-1}H^{k-1-j}EH_0^j.
H_0^kUse in source · 7 passages
Search aliases: H_0^k
Use in source · FINAL_MANUSCRIPT.tex, line 562
[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.
Exact mathematical passage
\boxed{ \rho\bigl((H^k-H_0^k)X\bigr) \leq (\mathcal B^k-\mathcal B_0^k)\rho(X). }
H_bUse in source · 4 passages
Search aliases: H_b
Use in source · FINAL_MANUSCRIPT.tex, line 343
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
H_b=G_bW_b^{1/2}
H_{b,I_r}Use in source · 2 passages
Search aliases: H_b,I_r
Use in source · FINAL_MANUSCRIPT.tex, line 354
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
IUse in source · 16 passages
Search aliases: I
Use in source · FINAL_MANUSCRIPT.tex, line 108
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
M_X=I-Q_XQ_X^\top, \qquad A=M_XGW^{1/2}, \qquad K=AA^\top.
I_1Use in source · 1 passage
Search aliases: I_1
Use in source · FINAL_MANUSCRIPT.tex, line 346
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
I_1,\ldots,I_R
I_RUse in source · 1 passage
Search aliases: I_R
Use in source · FINAL_MANUSCRIPT.tex, line 346
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
I_1,\ldots,I_R
I_rUse in source · 3 passages
Search aliases: I_r
Use in source · FINAL_MANUSCRIPT.tex, line 354
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
I_sUse in source · 1 passage
Search aliases: I_s
Use in source · FINAL_MANUSCRIPT.tex, line 371
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ K_{I_r,I_s} = \sum_b A_{b,I_r}A_{b,I_s}^\top. }
\infUse in source · 1 passage
Search aliases: inf
Use in source · FINAL_MANUSCRIPT.tex, line 1617
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat K_H(\theta)-\theta q\}, }
\inf_{0<2\theta(c+\rho)<1}Use in source · 1 passage
Search aliases: inf_0<2θ(c+ρ)<1 inf theta rho
Use in source · FINAL_MANUSCRIPT.tex, line 1617
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat K_H(\theta)-\theta q\}, }
\mathcal JUse in source · 4 passages
Search aliases: J mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 397
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\mathcal J
jUse in source · 13 passages
Search aliases: j
Use in source · FINAL_MANUSCRIPT.tex, line 100
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
J'Use in source · 1 passage
Search aliases: J'
Use in source · FINAL_MANUSCRIPT.tex, line 399
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
\mathcal J'Use in source · 2 passages
Search aliases: J' mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 398
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\mathcal J'
J_1Use in source · 1 passage
Search aliases: J_1
Use in source · FINAL_MANUSCRIPT.tex, line 149
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
\{1,\ldots,m\} = J_1\sqcup\cdots\sqcup J_B.
J_BUse in source · 1 passage
Search aliases: J_B
Use in source · FINAL_MANUSCRIPT.tex, line 149
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
\{1,\ldots,m\} = J_1\sqcup\cdots\sqcup J_B.
J_bUse in source · 1 passage
Search aliases: J_b
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
KUse in source · 21 passages
Search aliases: K
Use in source · FINAL_MANUSCRIPT.tex, line 61
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
A=M_XGW^{1/2}, \qquad K=AA^\top,
K_bUse in source · 1 passage
Search aliases: K_b
Use in source · FINAL_MANUSCRIPT.tex, line 211
[H] centering includegraphics[width=.92linewidth]figure_02_exact_block_kernel_assembly.png captionExact block-kernel assembly. The sample interaction kernel carried by each resident variant block is K_b=A_bA_b^top, and the full kernel is their exact sum. Streaming changes the order of evaluation, not the operator.
Exact mathematical passage
K_b=A_bA_b^\top
K_CUse in source · 1 passage
Search aliases: K_C
Use in source · FINAL_MANUSCRIPT.tex, line 1560
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\boxed{ K_H(\theta) \le K_C(\theta) + \theta\alpha t + \frac{\theta^2\alpha^2v} {1-2\theta\alpha\rho}. }
K_HUse in source · 2 passages
Search aliases: K_H
Use in source · FINAL_MANUSCRIPT.tex, line 1555
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
K_H(\theta) =-\frac12\log\det(I-2\theta H)
K_{I_r,I_s}Use in source · 1 passage
Search aliases: K_I_r,I_s
Use in source · FINAL_MANUSCRIPT.tex, line 371
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ K_{I_r,I_s} = \sum_b A_{b,I_r}A_{b,I_s}^\top. }
LUse in source · 2 passages
Search aliases: L
Use in source · FINAL_MANUSCRIPT.tex, line 2129
For an explicitly PSD polyphase operator H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86, and the largest norm observed outside the exact k-step light cone was 0.
Exact mathematical passage
H=L^\top L
lUse in source · 5 passages
Search aliases: l
Use in source · FINAL_MANUSCRIPT.tex, line 1001
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
T_{\rm serial}=\sum_{b=1}^{B}(t_b+c_b),
L^\topUse in source · 1 passage
Search aliases: L^top top
Use in source · FINAL_MANUSCRIPT.tex, line 2129
For an explicitly PSD polyphase operator H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86, and the largest norm observed outside the exact k-step light cone was 0.
Exact mathematical passage
H=L^\top L
\lceilUse in source · 1 passage
Search aliases: lceil
Use in source · FINAL_MANUSCRIPT.tex, line 1708
[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.
Exact mathematical passage
\boxed{ \left\lceil \max\left\{ \frac{t}{\rho}, \frac{t^2}{v} \right\} \right\rceil \le \operatorname{rank}F \le n-r. }
\logUse in source · 5 passages
Search aliases: log
Use in source · FINAL_MANUSCRIPT.tex, line 1555
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
K_H(\theta) =-\frac12\log\det(I-2\theta H)
\log_{10}Use in source · 2 passages
Search aliases: log_10 log
Use in source · FINAL_MANUSCRIPT.tex, line 1959
log_10p=-457.973916.
Exact mathematical passage
\boxed{\log_{10}p=-457.973916}.
\LongleftrightarrowUse in source · 1 passage
Search aliases: Longleftrightarrow
Use in source · FINAL_MANUSCRIPT.tex, line 441
Let P_h project onto variant phase h, and again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.
Exact mathematical passage
g\to h \quad\Longleftrightarrow\quad P_hHP_g\ne0.
\lVertUse in source · 15 passages
Search aliases: lVert
Use in source · FINAL_MANUSCRIPT.tex, line 503
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
MUse in source · 25 passages
Search aliases: M
Use in source · FINAL_MANUSCRIPT.tex, line 61
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
A=M_XGW^{1/2}, \qquad K=AA^\top,
M^{k-1}Use in source · 1 passage
Search aliases: M^k-1
Use in source · FINAL_MANUSCRIPT.tex, line 633
[Scalar threshold law] For any induced matrix norm, let M=max{lVert BrVert,lVert B_0rVert}. Then lVert B^k- B_0^krVert ≤ kM^k-1lVert RrVert. Hence weak discarded coupling produces a first-order error bound proportional to the discarded transfer norm and to the Krylov depth.
Exact mathematical passage
\boxed{ \lVert\mathcal B^k-\mathcal B_0^k\rVert \leq kM^{k-1}\lVert\mathcal R\rVert. }
M_{\rm block}Use in source · 2 passages
Search aliases: M_rm block rm
Use in source · FINAL_MANUSCRIPT.tex, line 1029
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm block}
M_{\rm double}Use in source · 1 passage
Search aliases: M_rm double rm
Use in source · FINAL_MANUSCRIPT.tex, line 1031
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm single}=M_{\rm fixed}+M_{\rm block}, \qquad \boxed{ M_{\rm double}=M_{\rm fixed}+2M_{\rm block}. }
M_{\rm fixed}Use in source · 2 passages
Search aliases: M_rm fixed rm
Use in source · FINAL_MANUSCRIPT.tex, line 1030
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm fixed}
M_{\rm full}Use in source · 1 passage
Search aliases: M_rm full rm
Use in source · FINAL_MANUSCRIPT.tex, line 2063
For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.
Exact mathematical passage
\boxed{ \frac{M_{\rm stream}}{M_{\rm full}} = O\!\left( \frac{c}{m} + \frac{s}{m} + \frac{p}{m} \right), }
M_{\rm single}Use in source · 1 passage
Search aliases: M_rm single rm
Use in source · FINAL_MANUSCRIPT.tex, line 1031
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm single}=M_{\rm fixed}+M_{\rm block}, \qquad \boxed{ M_{\rm double}=M_{\rm fixed}+2M_{\rm block}. }
M_{\rm stream}Use in source · 1 passage
Search aliases: M_rm stream rm
Use in source · FINAL_MANUSCRIPT.tex, line 2063
For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.
Exact mathematical passage
\boxed{ \frac{M_{\rm stream}}{M_{\rm full}} = O\!\left( \frac{c}{m} + \frac{s}{m} + \frac{p}{m} \right), }
M_XUse in source · 19 passages
Search aliases: M_X
Use in source · FINAL_MANUSCRIPT.tex, line 61
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
A=M_XGW^{1/2}, \qquad K=AA^\top,
M_X^2Use in source · 1 passage
Search aliases: M_X^2
Use in source · FINAL_MANUSCRIPT.tex, line 199
A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
M_X^\top=M_X=M_X^2
M_X^\topUse in source · 1 passage
Search aliases: M_X^top top
Use in source · FINAL_MANUSCRIPT.tex, line 199
A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
M_X^\top=M_X=M_X^2
NUse in source · 2 passages
Search aliases: N
Use in source · FINAL_MANUSCRIPT.tex, line 875
Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).
Exact mathematical passage
\beta_{(1)}\le\beta_{(2)}\le\cdots\le\beta_{(N)}.
nUse in source · 21 passages
Search aliases: n
Use in source · FINAL_MANUSCRIPT.tex, line 66
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
G\in\mathbb R^{n\times m}
OUse in source · 4 passages
Search aliases: O
Use in source · FINAL_MANUSCRIPT.tex, line 256
[Working memory independent of total variant count] If at most c variants are resident and the block-vector width is s, then the algebraic working set is O(nc+ns+cs+np), independent of total variant count m.
Exact mathematical passage
\boxed{ O(nc+ns+cs+np), }
oUse in source · 4 passages
Search aliases: o
Use in source · FINAL_MANUSCRIPT.tex, line 1029
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm block}
pUse in source · 21 passages
Search aliases: p
Use in source · FINAL_MANUSCRIPT.tex, line 106
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
Q_X\in\mathbb R^{n\times p}
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · FINAL_MANUSCRIPT.tex, line 501
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
P_1,\ldots,P_q
P_jUse in source · 1 passage
Search aliases: P_j
Use in source · FINAL_MANUSCRIPT.tex, line 458
Insert the projector resolution ∑_jP_j=I between successive factors of H. Every nonzero term in P_hH^kP_g corresponds to a phase path of length k. If no such path exists, every term vanishes.
Exact mathematical passage
\sum_jP_j=I
P_qUse in source · 1 passage
Search aliases: P_q
Use in source · FINAL_MANUSCRIPT.tex, line 501
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
P_1,\ldots,P_q
\PrUse in source · 3 passages
Search aliases: Pr
Use in source · FINAL_MANUSCRIPT.tex, line 1302
[Stochastic tail enclosure] For every threshold q, Pr(Q_-≥ q) ≤ Pr(Q≥ q) ≤ Pr(Q_+≥ q).
Exact mathematical passage
\boxed{ \Pr(Q_-\ge q) \le \Pr(Q\ge q) \le \Pr(Q_+\ge q). }
QUse in source · 20 passages
Search aliases: Q
Use in source · FINAL_MANUSCRIPT.tex, line 106
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
Q_X\in\mathbb R^{n\times p}
qUse in source · 9 passages
Search aliases: q
Use in source · FINAL_MANUSCRIPT.tex, line 501
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
P_1,\ldots,P_q
Q^\topUse in source · 2 passages
Search aliases: Q^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1101
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.
Exact mathematical passage
A=QR, \qquad Q^\top Q=I,
Q_+Use in source · 3 passages
Search aliases: Q_+
Use in source · FINAL_MANUSCRIPT.tex, line 1292
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Q_-=\sum_i\ell_iZ_i^2, \qquad Q=\sum_i\lambda_iZ_i^2, \qquad Q_+=\sum_i u_iZ_i^2.
Q_-Use in source · 3 passages
Search aliases: Q_-
Use in source · FINAL_MANUSCRIPT.tex, line 1292
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Q_-=\sum_i\ell_iZ_i^2, \qquad Q=\sum_i\lambda_iZ_i^2, \qquad Q_+=\sum_i u_iZ_i^2.
q_e^{(d)}Use in source · 1 passage
Search aliases: q_e^(d)
Use in source · FINAL_MANUSCRIPT.tex, line 832
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
q_e^{(k)}Use in source · 3 passages
Search aliases: q_e^(k)
Use in source · FINAL_MANUSCRIPT.tex, line 755
For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.
Exact mathematical passage
\boxed{ q_e^{(k)} = \sum_{j=0}^{k-1} \mathcal B^{k-1-j}\Delta_e\mathcal B^j\rho_0 }
q_e^{(p)}Use in source · 1 passage
Search aliases: q_e^(p)
Use in source · FINAL_MANUSCRIPT.tex, line 832
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
Q_rUse in source · 3 passages
Search aliases: Q_r
Use in source · FINAL_MANUSCRIPT.tex, line 1142
Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.
Exact mathematical passage
Q_r
Q_r^\topUse in source · 1 passage
Search aliases: Q_r^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1147
Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.
Exact mathematical passage
A=Q_rB, \qquad B=Q_r^\top A.
Q_XUse in source · 3 passages
Search aliases: Q_X
Use in source · FINAL_MANUSCRIPT.tex, line 106
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
Q_X\in\mathbb R^{n\times p}
Q_{X,I_r}Use in source · 1 passage
Search aliases: Q_X,I_r
Use in source · FINAL_MANUSCRIPT.tex, line 363
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ A_{b,I_r} = H_{b,I_r}-Q_{X,I_r}R_b. }
Q_{X,I_r}^\topUse in source · 1 passage
Search aliases: Q_X,I_r^top top
Use in source · FINAL_MANUSCRIPT.tex, line 354
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
Q_X^\topUse in source · 4 passages
Search aliases: Q_X^top top
Use in source · FINAL_MANUSCRIPT.tex, line 108
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
M_X=I-Q_XQ_X^\top, \qquad A=M_XGW^{1/2}, \qquad K=AA^\top.
Q_\perpUse in source · 2 passages
Search aliases: Q_⊥ perp
Use in source · FINAL_MANUSCRIPT.tex, line 1107
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp.
RUse in source · 20 passages
Search aliases: R
Use in source · FINAL_MANUSCRIPT.tex, line 346
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
I_1,\ldots,I_R
\mathbb RUse in source · 9 passages
Search aliases: R mathbb
Use in source · FINAL_MANUSCRIPT.tex, line 66
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
G\in\mathbb R^{n\times m}
\mathcal RUse in source · 4 passages
Search aliases: R mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 547
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.
Exact mathematical passage
(\mathcal B_0)_{hg}= \begin{cases} \mathcal B_{hg},&(h,g)\in\mathcal E_0,\\ 0,&\text{otherwise}, \end{cases} \qquad \mathcal R=\mathcal B-\mathcal B_0.
rUse in source · 34 passages
Search aliases: r
Use in source · FINAL_MANUSCRIPT.tex, line 354
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
R^2Use in source · 1 passage
Search aliases: R^2
Use in source · FINAL_MANUSCRIPT.tex, line 1517
This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the residual spectrum is; the third is its largest possible coefficient.
Exact mathematical passage
t=\operatorname{tr}R, \qquad v=\operatorname{tr}(R^2), \qquad \rho=\|R\|_2.
r^2Use in source · 2 passages
Search aliases: r^2
Use in source · FINAL_MANUSCRIPT.tex, line 1856
[H] centering captionRealistic rare-variant simulation scenarios. Adjacent r^2 is computed within the simulated LD blocks after variant standardization. lrrrrr scenario · n · m · rank · median adjacent r^2 · effective rank
Exact mathematical passage
r^2
\mathbb R^{m\times s}Use in source · 2 passages
Search aliases: R^m× s mathbb times
Use in source · FINAL_MANUSCRIPT.tex, line 266
Let V∈ R^m× s be partitioned conformally: V= V_1
Exact mathematical passage
V\in\mathbb R^{m\times s}
\mathbb R^{n\times m}Use in source · 2 passages
Search aliases: R^n× m mathbb times
Use in source · FINAL_MANUSCRIPT.tex, line 66
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
G\in\mathbb R^{n\times m}
\mathbb R^{n\times p}Use in source · 1 passage
Search aliases: R^n× p mathbb times
Use in source · FINAL_MANUSCRIPT.tex, line 106
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
Q_X\in\mathbb R^{n\times p}
\mathbb R^{n\times r}Use in source · 1 passage
Search aliases: R^n× r mathbb times
Use in source · FINAL_MANUSCRIPT.tex, line 1481
Let U∈ R^n× r have orthonormal columns and define B=U^top A, F=(I-UU^top)A. Then A=UB+F, U^top F=0.
Exact mathematical passage
U\in\mathbb R^{n\times r}
\mathbb R^{n\times s}Use in source · 3 passages
Search aliases: R^n× s mathbb times
Use in source · FINAL_MANUSCRIPT.tex, line 218
[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.
Exact mathematical passage
V\in\mathbb R^{n\times s}
r_1Use in source · 1 passage
Search aliases: r_1
Use in source · FINAL_MANUSCRIPT.tex, line 1698
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
r_1,\ldots,r_s
R_bUse in source · 4 passages
Search aliases: R_b
Use in source · FINAL_MANUSCRIPT.tex, line 350
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
R_b=Q_X^\top H_b.
r_gUse in source · 1 passage
Search aliases: r_g
Use in source · FINAL_MANUSCRIPT.tex, line 1217
For any vector x, let r_g= P_gx _2. Then P_hEx _2 ≤ ∑_g P_hEP_g _2 P_gx _2 = ( C r)_h. Orthogonality of the phase decomposition gives Ex _2^2 = ∑_h P_hEx _2^2 ≤ C r _2^2 ≤ C _2^2 r _2^2 = C _2^2 x _2^2. Taking the supremum over unit vectors proves the claim.
Exact mathematical passage
r_g=\|P_gx\|_2.
R_hUse in source · 3 passages
Search aliases: R_h
Use in source · FINAL_MANUSCRIPT.tex, line 1052
Let R be a newly generated Krylov residual and R_h=P_hR.
Exact mathematical passage
R_h=P_hR.
r_hUse in source · 3 passages
Search aliases: r_h
Use in source · FINAL_MANUSCRIPT.tex, line 1058
[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.
Exact mathematical passage
r_h=\operatorname{rank}(R_h).
r_{hg}Use in source · 1 passage
Search aliases: r_hg
Use in source · FINAL_MANUSCRIPT.tex, line 1084
If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).
Exact mathematical passage
r_{hg}
r_iUse in source · 2 passages
Search aliases: r_i
Use in source · FINAL_MANUSCRIPT.tex, line 1699
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
t=\sum_i r_i, \qquad v=\sum_i r_i^2, \qquad r_i\le\rho.
r_i^2Use in source · 2 passages
Search aliases: r_i^2
Use in source · FINAL_MANUSCRIPT.tex, line 1699
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
t=\sum_i r_i, \qquad v=\sum_i r_i^2, \qquad r_i\le\rho.
r_sUse in source · 1 passage
Search aliases: r_s
Use in source · FINAL_MANUSCRIPT.tex, line 1698
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
r_1,\ldots,r_s
R_\perpUse in source · 2 passages
Search aliases: R_⊥ perp
Use in source · FINAL_MANUSCRIPT.tex, line 1107
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp.
\operatorname{range}Use in source · 3 passages
Search aliases: range operatorname
Use in source · FINAL_MANUSCRIPT.tex, line 1061
[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.
Exact mathematical passage
\operatorname{range}(R_h)
\operatorname{rank}Use in source · 3 passages
Search aliases: rank operatorname
Use in source · FINAL_MANUSCRIPT.tex, line 1058
[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.
Exact mathematical passage
r_h=\operatorname{rank}(R_h).
\rceilUse in source · 1 passage
Search aliases: rceil
Use in source · FINAL_MANUSCRIPT.tex, line 1708
[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.
Exact mathematical passage
\boxed{ \left\lceil \max\left\{ \frac{t}{\rho}, \frac{t^2}{v} \right\} \right\rceil \le \operatorname{rank}F \le n-r. }
\rmUse in source · 9 passages
Search aliases: rm
Use in source · FINAL_MANUSCRIPT.tex, line 1001
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
T_{\rm serial}=\sum_{b=1}^{B}(t_b+c_b),
\rVertUse in source · 15 passages
Search aliases: rVert
Use in source · FINAL_MANUSCRIPT.tex, line 503
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
\rVert_2Use in source · 9 passages
Search aliases: rVert_2 rVert
Use in source · FINAL_MANUSCRIPT.tex, line 507
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\mathcal B_{hg}=\lVert P_hHP_g\rVert_2.
\rVert_FUse in source · 6 passages
Search aliases: rVert_F rVert
Use in source · FINAL_MANUSCRIPT.tex, line 503
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
\rVert_F^2Use in source · 1 passage
Search aliases: rVert_F^2 rVert
Use in source · FINAL_MANUSCRIPT.tex, line 602
ing Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
\lVert Y\rVert_F^2=\sum_h\lVert P_hY\rVert_F^2,
sUse in source · 22 passages
Search aliases: s
Use in source · FINAL_MANUSCRIPT.tex, line 218
[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.
Exact mathematical passage
V\in\mathbb R^{n\times s}
S'Use in source · 5 passages
Search aliases: S'
Use in source · FINAL_MANUSCRIPT.tex, line 2229
[Storage-phase commutation theorem] Let A=M_XGW^1/2, H=A^top A, and let S and S' be any two exact storage refinements that respect the same structural phase partition. Then: [label=(roman*)] item the exact operator H is identical under S and S'; item every phase coupling P_hHP_g is identical under S and S'; item the exact phase graph and every k-step polyphase light cone are identical under S and S'; item any exact matrix-free Krylov method applied through streamed products produces the same Kry…
Exact mathematical passage
S'
\succeqUse in source · 2 passages
Search aliases: succeq
Use in source · FINAL_MANUSCRIPT.tex, line 1251
Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.
Exact mathematical passage
H\succeq0
TUse in source · 9 passages
Search aliases: T
Use in source · FINAL_MANUSCRIPT.tex, line 244
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
T_b=G_b^\top U, \qquad Y\leftarrow Y+G_b(W_bT_b).
tUse in source · 23 passages
Search aliases: t
Use in source · FINAL_MANUSCRIPT.tex, line 388
With sample tile height t and resident variant width c, the dominant working storage is O(tc+pc+t^2).
Exact mathematical passage
t
T^2Use in source · 1 passage
Search aliases: T^2
Use in source · FINAL_MANUSCRIPT.tex, line 1594
Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u) and a geometric bound on all terms of degree at least two. Summing over the eigenvalues of T proves the result.
Exact mathematical passage
\operatorname{tr}T\le\alpha t, \qquad \operatorname{tr}(T^2)\le\alpha^2v, \qquad \|T\|_2\le\alpha\rho.
t^2Use in source · 4 passages
Search aliases: t^2
Use in source · FINAL_MANUSCRIPT.tex, line 390
With sample tile height t and resident variant width c, the dominant working storage is O(tc+pc+t^2).
Exact mathematical passage
O(tc+pc+t^2).
t^dUse in source · 1 passage
Search aliases: t^d
Use in source · FINAL_MANUSCRIPT.tex, line 828
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
t^jUse in source · 1 passage
Search aliases: t^j
Use in source · FINAL_MANUSCRIPT.tex, line 647
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
t_1Use in source · 1 passage
Search aliases: t_1
Use in source · FINAL_MANUSCRIPT.tex, line 1005
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
T_bUse in source · 1 passage
Search aliases: T_b
Use in source · FINAL_MANUSCRIPT.tex, line 244
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
T_b=G_b^\top U, \qquad Y\leftarrow Y+G_b(W_bT_b).
t_bUse in source · 3 passages
Search aliases: t_b
Use in source · FINAL_MANUSCRIPT.tex, line 995
The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.
Exact mathematical passage
t_b
t_{b+1}Use in source · 1 passage
Search aliases: t_b+1
Use in source · FINAL_MANUSCRIPT.tex, line 1005
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
T_{\rm pipe}Use in source · 2 passages
Search aliases: T_rm pipe rm
Use in source · FINAL_MANUSCRIPT.tex, line 1005
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
T_{\rm serial}Use in source · 1 passage
Search aliases: T_rm serial rm
Use in source · FINAL_MANUSCRIPT.tex, line 1001
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
T_{\rm serial}=\sum_{b=1}^{B}(t_b+c_b),
\operatorname{tr}Use in source · 2 passages
Search aliases: tr operatorname
Use in source · FINAL_MANUSCRIPT.tex, line 1517
This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the residual spectrum is; the third is its largest possible coefficient.
Exact mathematical passage
t=\operatorname{tr}R, \qquad v=\operatorname{tr}(R^2), \qquad \rho=\|R\|_2.
UUse in source · 30 passages
Search aliases: U
Use in source · FINAL_MANUSCRIPT.tex, line 241
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
U=M_XV
uUse in source · 7 passages
Search aliases: u
Use in source · FINAL_MANUSCRIPT.tex, line 1031
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm single}=M_{\rm fixed}+M_{\rm block}, \qquad \boxed{ M_{\rm double}=M_{\rm fixed}+2M_{\rm block}. }
U^\topUse in source · 6 passages
Search aliases: U^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1388
There is a second route that does not use the pruned operator as the final spectral approximation. Let U be any orthonormal sample-space basis, possibly generated cheaply from a pruned or approximate Krylov process, and define B=U^top A, F=(I-UU^top)A, δ= F _2.
Exact mathematical passage
B=U^\top A, \qquad F=(I-UU^\top)A, \qquad \delta=\|F\|_2.
U_{hg}Use in source · 2 passages
Search aliases: U_hg
Use in source · FINAL_MANUSCRIPT.tex, line 1079
If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).
Exact mathematical passage
P_hHP_g = U_{hg}V_{hg}^\top
u_iUse in source · 3 passages
Search aliases: u_i
Use in source · FINAL_MANUSCRIPT.tex, line 1271
[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.
Exact mathematical passage
\boxed{ \ell_i = \max\{0,\mu_i-\varepsilon\} \le \lambda_i \le u_i = \max\{0,\mu_i+\varepsilon\}. }
VUse in source · 22 passages
Search aliases: V
Use in source · FINAL_MANUSCRIPT.tex, line 218
[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.
Exact mathematical passage
V\in\mathbb R^{n\times s}
vUse in source · 9 passages
Search aliases: v
Use in source · FINAL_MANUSCRIPT.tex, line 1517
This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the residual spectrum is; the third is its largest possible coefficient.
Exact mathematical passage
t=\operatorname{tr}R, \qquad v=\operatorname{tr}(R^2), \qquad \rho=\|R\|_2.
V_1Use in source · 1 passage
Search aliases: V_1
Use in source · FINAL_MANUSCRIPT.tex, line 267
Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .
Exact mathematical passage
V= \begin{bmatrix} V_1\\ \vdots\\ V_B \end{bmatrix}.
V_BUse in source · 1 passage
Search aliases: V_B
Use in source · FINAL_MANUSCRIPT.tex, line 267
Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .
Exact mathematical passage
V= \begin{bmatrix} V_1\\ \vdots\\ V_B \end{bmatrix}.
V_bUse in source · 1 passage
Search aliases: V_b
Use in source · FINAL_MANUSCRIPT.tex, line 277
[Streamed AV] AV = M_X ∑_b=1^B G_bW_b^1/2V_b.
Exact mathematical passage
\boxed{ AV = M_X \sum_{b=1}^B G_bW_b^{1/2}V_b. }
V_{hg}^\topUse in source · 2 passages
Search aliases: V_hg^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1079
If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).
Exact mathematical passage
P_hHP_g = U_{hg}V_{hg}^\top
\vdotsUse in source · 2 passages
Search aliases: vdots
Use in source · FINAL_MANUSCRIPT.tex, line 187
Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
[A_1\ \cdots\ A_B] \begin{bmatrix} A_1^\top\\ \vdots\\ A_B^\top \end{bmatrix} = \sum_bA_bA_b^\top.
WUse in source · 23 passages
Search aliases: W
Use in source · FINAL_MANUSCRIPT.tex, line 61
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
A=M_XGW^{1/2}, \qquad K=AA^\top,
wUse in source · 1 passage
Search aliases: w
Use in source · FINAL_MANUSCRIPT.tex, line 100
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
W^{1/2}Use in source · 5 passages
Search aliases: W^1/2
Use in source · FINAL_MANUSCRIPT.tex, line 61
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
A=M_XGW^{1/2}, \qquad K=AA^\top,
w_1Use in source · 1 passage
Search aliases: w_1
Use in source · FINAL_MANUSCRIPT.tex, line 100
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
W_bUse in source · 5 passages
Search aliases: W_b
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
W_b^{1/2}Use in source · 7 passages
Search aliases: W_b^1/2
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
w_jUse in source · 1 passage
Search aliases: w_j
Use in source · FINAL_MANUSCRIPT.tex, line 100
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
W_{J_b,J_b}Use in source · 1 passage
Search aliases: W_J_b,J_b
Use in source · FINAL_MANUSCRIPT.tex, line 155
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
w_mUse in source · 1 passage
Search aliases: w_m
Use in source · FINAL_MANUSCRIPT.tex, line 100
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
\widehat KUse in source · 2 passages
Search aliases: widehat K widehat
Use in source · FINAL_MANUSCRIPT.tex, line 1617
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat K_H(\theta)-\theta q\}, }
\widehat K_HUse in source · 2 passages
Search aliases: widehat K_H widehat
Use in source · FINAL_MANUSCRIPT.tex, line 1617
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat K_H(\theta)-\theta q\}, }
XUse in source · 48 passages
Search aliases: X
Use in source · FINAL_MANUSCRIPT.tex, line 61
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
A=M_XGW^{1/2}, \qquad K=AA^\top,
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · FINAL_MANUSCRIPT.tex, line 1602
ginproof Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u) and a geometric bound on all terms of degree at least two. Summing over the eigenvalues of T proves the result.
Exact mathematical passage
-\frac12\log(1-2\theta x) \le \theta x + \frac{\theta^2x^2} {1-2\theta\alpha\rho}.
YUse in source · 6 passages
Search aliases: Y
Use in source · FINAL_MANUSCRIPT.tex, line 242
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
Y=0
ZUse in source · 7 passages
Search aliases: Z
Use in source · FINAL_MANUSCRIPT.tex, line 1291
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Z_i
Z^\topUse in source · 3 passages
Search aliases: Z^top top
Use in source · FINAL_MANUSCRIPT.tex, line 1616
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.
Exact mathematical passage
Q=Z^\top HZ
Z_iUse in source · 2 passages
Search aliases: Z_i
Use in source · FINAL_MANUSCRIPT.tex, line 1291
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Z_i
Z_i^2Use in source · 1 passage
Search aliases: Z_i^2
Use in source · FINAL_MANUSCRIPT.tex, line 1292
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Q_-=\sum_i\ell_iZ_i^2, \qquad Q=\sum_i\lambda_iZ_i^2, \qquad Q_+=\sum_i u_iZ_i^2.
\timesUse in source · 34 passages
Search aliases: × times
Use in source · FINAL_MANUSCRIPT.tex, line 66
Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model covariates. When m is large, the direct factor A may exceed device memory even though the final statistic requires only a spectrum or repeated operat…
Exact mathematical passage
G\in\mathbb R^{n\times m}
\alphaUse in source · 5 passages
Search aliases: α alpha
Use in source · FINAL_MANUSCRIPT.tex, line 1549
Set C=B^top B, H=C+R, c=λ_max(C). For 0<2θ(c+ρ)<1, put α=(1-2θ c)^-1.
Exact mathematical passage
\alpha=(1-2\theta c)^{-1}.
\alpha^2Use in source · 2 passages
Search aliases: α^2 alpha
Use in source · FINAL_MANUSCRIPT.tex, line 1560
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\boxed{ K_H(\theta) \le K_C(\theta) + \theta\alpha t + \frac{\theta^2\alpha^2v} {1-2\theta\alpha\rho}. }
\beta_{(1)}Use in source · 1 passage
Search aliases: β_(1) beta
Use in source · FINAL_MANUSCRIPT.tex, line 875
Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).
Exact mathematical passage
\beta_{(1)}\le\beta_{(2)}\le\cdots\le\beta_{(N)}.
\beta_{(2)}Use in source · 1 passage
Search aliases: β_(2) beta
Use in source · FINAL_MANUSCRIPT.tex, line 875
Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).
Exact mathematical passage
\beta_{(1)}\le\beta_{(2)}\le\cdots\le\beta_{(N)}.
\beta_{(i)}Use in source · 2 passages
Search aliases: β_(i) beta
Use in source · FINAL_MANUSCRIPT.tex, line 882
[Cardinality-optimal certified pruning] Let r be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.
Exact mathematical passage
\sum_{i=1}^{r}\beta_{(i)}\le\tau.
\beta_{(N)}Use in source · 1 passage
Search aliases: β_(N) beta
Use in source · FINAL_MANUSCRIPT.tex, line 875
Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).
Exact mathematical passage
\beta_{(1)}\le\beta_{(2)}\le\cdots\le\beta_{(N)}.
\beta_e^{(p)}Use in source · 2 passages
Search aliases: β_e^(p) beta
Use in source · FINAL_MANUSCRIPT.tex, line 832
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
\beta_{hg}^{(5)}Use in source · 1 passage
Search aliases: β_hg^(5) beta
Use in source · FINAL_MANUSCRIPT.tex, line 958
[H] centering includegraphics[width=.82linewidth]figure_13_edge_risk_vs_block_norm.png captionRaw phase-block magnitude and finite-depth query risk are different quantities. The edge score β_hg^(5) incorporates the start profile and every depth-five transfer path through the phase graph; the block norm B_hg does not.
Exact mathematical passage
\beta_{hg}^{(5)}
\varepsilonUse in source · 3 passages
Search aliases: ε varepsilon
Use in source · FINAL_MANUSCRIPT.tex, line 1261
Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.
Exact mathematical passage
\|H-H_0\|_2\le\varepsilon,
\thetaUse in source · 9 passages
Search aliases: θ theta
Use in source · FINAL_MANUSCRIPT.tex, line 1548
Set C=B^top B, H=C+R, c=λ_max(C). For 0<2θ(c+ρ)<1, put α=(1-2θ c)^-1.
Exact mathematical passage
0<2\theta(c+\rho)<1
\theta^2Use in source · 2 passages
Search aliases: θ^2 theta
Use in source · FINAL_MANUSCRIPT.tex, line 1560
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\boxed{ K_H(\theta) \le K_C(\theta) + \theta\alpha t + \frac{\theta^2\alpha^2v} {1-2\theta\alpha\rho}. }
\lambda_jUse in source · 2 passages
Search aliases: λ_j lambda
Use in source · FINAL_MANUSCRIPT.tex, line 2015
The matrix packets above operate before eigendecomposition. They should be distinguished from the spectral packets used after the eigenvalues λ_j are known.
Exact mathematical passage
\lambda_j
\lambda_{\max}Use in source · 1 passage
Search aliases: λ_max lambda max
Use in source · FINAL_MANUSCRIPT.tex, line 1541
Set C=B^top B, H=C+R, c=λ_max(C). For 0<2θ(c+ρ)<1, put α=(1-2θ c)^-1.
Exact mathematical passage
C=B^\top B, \qquad H=C+R, \qquad c=\lambda_{\max}(C).
\muUse in source · 3 passages
Search aliases: μ mu
Use in source · FINAL_MANUSCRIPT.tex, line 1255
Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.
Exact mathematical passage
\lambda_i \qquad\text{and}\qquad \mu_i.
\mu_iUse in source · 3 passages
Search aliases: μ_i mu
Use in source · FINAL_MANUSCRIPT.tex, line 1255
Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.
Exact mathematical passage
\lambda_i \qquad\text{and}\qquad \mu_i.
\rhoUse in source · 35 passages
Search aliases: ρ rho
Use in source · FINAL_MANUSCRIPT.tex, line 503
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
\rho_0Use in source · 9 passages
Search aliases: ρ_0 rho
Use in source · FINAL_MANUSCRIPT.tex, line 753
For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.
Exact mathematical passage
\rho_0=\rho(X)
\tau_KUse in source · 1 passage
Search aliases: τ_K tau
Use in source · FINAL_MANUSCRIPT.tex, line 1338
A production schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.
Exact mathematical passage
\boxed{ \sum_{e\in\mathcal O}\beta_e^{(k)} \le \tau_K, \qquad \|\mathcal C_{\mathcal O}\|_2 \le \tau_\lambda. }
\tau_\lambdaUse in source · 1 passage
Search aliases: τ_λ tau lambda
Use in source · FINAL_MANUSCRIPT.tex, line 1338
A production schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.
Exact mathematical passage
\boxed{ \sum_{e\in\mathcal O}\beta_e^{(k)} \le \tau_K, \qquad \|\mathcal C_{\mathcal O}\|_2 \le \tau_\lambda. }
\chiUse in source · 5 passages
Search aliases: χ chi
Use in source · FINAL_MANUSCRIPT.tex, line 82
er composes with an exact polyphase light-cone theorem, minimal phase-local rank allocation, low-rank interface messages, and exact QR-core updates. When phase couplings are weak rather than zero, a nonnegative transfer matrix gives certified phasewise leakage bounds for operator powers and complete Krylov polynomials. A dual query/spectral budget propagates pruning uncertainty to generalized-χ^2 tail brackets, while residual trace, square-trace, and spectral-cap invariants supply substantially …
Exact mathematical passage
\chi^2
\chi^2Use in source · 5 passages
Search aliases: χ^2 chi
Use in source · FINAL_MANUSCRIPT.tex, line 82
er composes with an exact polyphase light-cone theorem, minimal phase-local rank allocation, low-rank interface messages, and exact QR-core updates. When phase couplings are weak rather than zero, a nonnegative transfer matrix gives certified phasewise leakage bounds for operator powers and complete Krylov polynomials. A dual query/spectral budget propagates pruning uncertainty to generalized-χ^2 tail brackets, while residual trace, square-trace, and spectral-cap invariants supply substantially …
Exact mathematical passage
\chi^2
\ellUse in source · 4 passages
Search aliases: ℓ ell
Use in source · FINAL_MANUSCRIPT.tex, line 1271
[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.
Exact mathematical passage
\boxed{ \ell_i = \max\{0,\mu_i-\varepsilon\} \le \lambda_i \le u_i = \max\{0,\mu_i+\varepsilon\}. }
\ell_2Use in source · 1 passage
Search aliases: ℓ_2 ell
Use in source · FINAL_MANUSCRIPT.tex, line 2142
The QR-core update reproduced the full singular spectrum with relative ℓ_2 discrepancy 5.89×10^-16.
Exact mathematical passage
\ell_2
\ell_iUse in source · 3 passages
Search aliases: ℓ_i ell
Use in source · FINAL_MANUSCRIPT.tex, line 1271
[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.
Exact mathematical passage
\boxed{ \ell_i = \max\{0,\mu_i-\varepsilon\} \le \lambda_i \le u_i = \max\{0,\mu_i+\varepsilon\}. }
\leftarrowUse in source · 1 passage
Search aliases: ← leftarrow
Use in source · FINAL_MANUSCRIPT.tex, line 244
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
T_b=G_b^\top U, \qquad Y\leftarrow Y+G_b(W_bT_b).
\toUse in source · 5 passages
Search aliases: → to
Use in source · FINAL_MANUSCRIPT.tex, line 441
Let P_h project onto variant phase h, and again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.
Exact mathematical passage
g\to h \quad\Longleftrightarrow\quad P_hHP_g\ne0.
\notinUse in source · 1 passage
Search aliases: ∉ notin
Use in source · FINAL_MANUSCRIPT.tex, line 2276
For every phase h∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Therefore blocks outside the active structural frontier contribute identically zero to the requested iterate and may be omitted.
Exact mathematical passage
h\notin F_k
\sum_{(h,g)\in\mathcal E_0}Use in source · 1 passage
Search aliases: ∑_(h,g)∈ E_0 sum in mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 543
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,
Exact mathematical passage
H_0=\sum_{(h,g)\in\mathcal E_0}P_hHP_g.
\sum_bUse in source · 2 passages
Search aliases: ∑_b sum
Use in source · FINAL_MANUSCRIPT.tex, line 187
Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
[A_1\ \cdots\ A_B] \begin{bmatrix} A_1^\top\\ \vdots\\ A_B^\top \end{bmatrix} = \sum_bA_bA_b^\top.
\sum_{b=1}^BUse in source · 3 passages
Search aliases: ∑_b=1^B sum
Use in source · FINAL_MANUSCRIPT.tex, line 168
[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.
Exact mathematical passage
\boxed{ K = AA^\top = \sum_{b=1}^B A_bA_b^\top = M_X \left( \sum_{b=1}^B G_bW_bG_b^\top \right) M_X. }
\sum_{b=1}^{B}Use in source · 1 passage
Search aliases: ∑_b=1^B sum
Use in source · FINAL_MANUSCRIPT.tex, line 1001
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
T_{\rm serial}=\sum_{b=1}^{B}(t_b+c_b),
\sum_{b=1}^{B-1}Use in source · 1 passage
Search aliases: ∑_b=1^B-1 sum
Use in source · FINAL_MANUSCRIPT.tex, line 1005
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
\sum_{d=0}^{D}Use in source · 1 passage
Search aliases: ∑_d=0^D sum
Use in source · FINAL_MANUSCRIPT.tex, line 828
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
\sum_{d=1}^{D}Use in source · 1 passage
Search aliases: ∑_d=1^D sum
Use in source · FINAL_MANUSCRIPT.tex, line 832
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
\sum_gUse in source · 3 passages
Search aliases: ∑_g sum
Use in source · FINAL_MANUSCRIPT.tex, line 530
Using ∑_gP_g=I and submultiplicativity, lVert P_hHXrVert_F · =lVert∑_gP_hHP_gXrVert_F
Exact mathematical passage
\sum_gP_g=I
\sum_hUse in source · 2 passages
Search aliases: ∑_h sum
Use in source · FINAL_MANUSCRIPT.tex, line 602
ing Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
\lVert Y\rVert_F^2=\sum_h\lVert P_hY\rVert_F^2,
\sum_iUse in source · 2 passages
Search aliases: ∑_i sum
Use in source · FINAL_MANUSCRIPT.tex, line 1292
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Q_-=\sum_i\ell_iZ_i^2, \qquad Q=\sum_i\lambda_iZ_i^2, \qquad Q_+=\sum_i u_iZ_i^2.
\sum_{i=1}^{r}Use in source · 1 passage
Search aliases: ∑_i=1^r sum
Use in source · FINAL_MANUSCRIPT.tex, line 882
[Cardinality-optimal certified pruning] Let r be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.
Exact mathematical passage
\sum_{i=1}^{r}\beta_{(i)}\le\tau.
\sum_{i=1}^{r+1}Use in source · 1 passage
Search aliases: ∑_i=1^r+1 sum
Use in source · FINAL_MANUSCRIPT.tex, line 892
Among all subsets of cardinality r+1, the smallest possible sum of risks is ∑_i=1^r+1β_(i). By maximality of r, that sum exceeds τ. Hence no set of r+1 omitted pairs is feasible. The prefix of length r is feasible by construction.
Exact mathematical passage
\sum_{i=1}^{r+1}\beta_{(i)}.
\sum_{i=1}^sUse in source · 1 passage
Search aliases: ∑_i=1^s sum
Use in source · FINAL_MANUSCRIPT.tex, line 1727
The inequality t≤ sρ gives s≥ t/ρ. Cauchy--Schwarz gives t^2 = (∑_i=1^s r_i)^2 ≤ s∑_i=1^s r_i^2 =sv, so s≥ t^2/v. Finally, every column of F lies in range(U)^⊥, whose dimension is n-r.
Exact mathematical passage
t^2 = \left(\sum_{i=1}^s r_i\right)^2 \le s\sum_{i=1}^s r_i^2 =sv,
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · FINAL_MANUSCRIPT.tex, line 458
Insert the projector resolution ∑_jP_j=I between successive factors of H. Every nonzero term in P_hH^kP_g corresponds to a phase path of length k. If no such path exists, every term vanishes.
Exact mathematical passage
\sum_jP_j=I
\sum_{J'\in\mathcal J'}Use in source · 1 passage
Search aliases: ∑_J'∈ J' sum in mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 399
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
\sum_{j=0}^dUse in source · 1 passage
Search aliases: ∑_j=0^d sum
Use in source · FINAL_MANUSCRIPT.tex, line 647
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
\sum_{j=0}^{k-1}Use in source · 5 passages
Search aliases: ∑_j=0^k-1 sum
Use in source · FINAL_MANUSCRIPT.tex, line 581
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
H^k-H_0^k = \sum_{j=0}^{k-1}H^{k-1-j}EH_0^j.
\sum_{j=1}^dUse in source · 1 passage
Search aliases: ∑_j=1^d sum
Use in source · FINAL_MANUSCRIPT.tex, line 651
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
\boxed{ \rho\bigl((p(H)-p(H_0))X\bigr) \leq \sum_{j=1}^d |a_j|(\mathcal B^j-\mathcal B_0^j)\rho(X). }
\sum_{J\in\mathcal J}Use in source · 1 passage
Search aliases: ∑_J∈ J sum in mathcal
Use in source · FINAL_MANUSCRIPT.tex, line 399
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
\sum_rUse in source · 1 passage
Search aliases: ∑_r sum
Use in source · FINAL_MANUSCRIPT.tex, line 354
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
\circUse in source · 2 passages
Search aliases: ∘ circ
Use in source · FINAL_MANUSCRIPT.tex, line 121
The central implementation difficulty is not the algebraic size of the model but the mismatch between the total factor size n× m and the working memory of an accelerator. The key point of this manuscript is that several decompositions act on different mathematical coordinates and therefore compose: storage blocks ∘ structural phases ∘ Krylov/range compression ∘ spectral tail packets.
Exact mathematical passage
\boxed{ \text{storage blocks} \circ \text{structural phases} \circ \text{Krylov/range compression} \circ \text{spectral tail packets}. }
\simUse in source · 1 passage
Search aliases: ∼ sim
Use in source · FINAL_MANUSCRIPT.tex, line 1616
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.
Exact mathematical passage
Z\sim N(0,I)
\neUse in source · 1 passage
Search aliases: ≠ ne
Use in source · FINAL_MANUSCRIPT.tex, line 441
Let P_h project onto variant phase h, and again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.
Exact mathematical passage
g\to h \quad\Longleftrightarrow\quad P_hHP_g\ne0.
\leqUse in source · 10 passages
Search aliases: ≤ leq
Use in source · FINAL_MANUSCRIPT.tex, line 516
[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).
Exact mathematical passage
\boxed{ \rho(HX)\leq \mathcal B\rho(X) }
\geqUse in source · 2 passages
Search aliases: ≥ geq
Use in source · FINAL_MANUSCRIPT.tex, line 521
[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).
Exact mathematical passage
k\geq0
\perpUse in source · 3 passages
Search aliases: ⊥ perp
Use in source · FINAL_MANUSCRIPT.tex, line 1107
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp.
\longrightarrowUse in source · 3 passages
Search aliases: ⟶ longrightarrow
Use in source · FINAL_MANUSCRIPT.tex, line 484
This is the key composition law: mathematical locality ⟶ active phases ⟶ active storage blocks.
Exact mathematical passage
\boxed{ \text{mathematical locality} \longrightarrow \text{active phases} \longrightarrow \text{active storage blocks}. }
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