Paper VI: Memory modes and controlled compression

E15 Memory modes and controlled compression

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Stable hidden modes may carry polynomial factors when a Jordan block is present. A spectral gap controls eventual decay, while nonnormal coupling can produce substantial intermediate memory. The compression certificate below is optimal for its declared primary-block selection problem.

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A two-coordinate Jordan hidden block has an exponential decay factor and a polynomial factor. The nilpotent polynomial is checked exactly at the displayed rational times. The exponential curve is a plotted illustration of the formula.

Finite hidden systems produce rational self energies and finite modal memory. This is the state space viewpoint familiar from linear systems theory [Kailath1980]. The same formulas also underlie the Mori Zwanzig representation of reduced dynamics [Zwanzig1961], [ChorinHaldKupferman2000].

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Theorem E15.1: Polynomial exponential memory of a finite hidden system

Let \(X_Q\) be a finite dimensional complex space. Let \(\mu_1,\ldots,\mu_r\) be the distinct eigenvalues of \(A_{QQ}\), let \(Q_j\) be the corresponding primary spectral projections, and set

\[ N_j=(A_{QQ}-\mu_jI)Q_j. \]

Let \(\nu_j\) be the least positive integer for which \(N_j^{\nu_j}=0\), and define

\[ B_{j,k} = A_{PQ}N_j^kQ_jA_{QP} \qquad (0\leq k<\nu_j). \]

Then

\[ {\mathcal K}(t) = \sum_{j=1}^r {\mathrm e}^{\mu_jt} \sum_{k=0}^{\nu_j-1} \frac{t^k}{k!}B_{j,k} \]

and

\[ \Sigma_P(z) = \sum_{j=1}^r \sum_{k=0}^{\nu_j-1} \frac{B_{j,k}}{(z-\mu_j)^{k+1}}. \]

Suppose \(\operatorname{Re}\mu_j<0\) for every \(j\). If \(J\subseteq\{1,\ldots,r\}\) and \({\mathcal K}_J\) retains the complete primary blocks indexed by \(J\), then

\[ {\left\lVert{\mathcal K}-{\mathcal K}_J\right\rVert}_{L^1(0,\infty)} \leq \sum_{j\notin J} \sum_{k=0}^{\nu_j-1} \frac{{\left\lVert B_{j,k}\right\rVert}} {(-\operatorname{Re}\mu_j)^{k+1}}. \]

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Proof of Theorem E15.1

The primary decomposition gives

\[ A_{QQ}Q_j=\mu_jQ_j+N_j \]

with commuting scalar and nilpotent parts on \(Q_jX_Q\). Therefore

\[ {\mathrm e}^{tA_{QQ}}Q_j = {\mathrm e}^{\mu_jt} \sum_{k=0}^{\nu_j-1} \frac{t^k}{k!}N_j^kQ_j. \]

Multiplication by \(A_{PQ}\) and \(A_{QP}\) gives the memory formula. On the \(j\)th primary space, the nilpotent resolvent expansion is

\[ (zI-A_{QQ})^{-1}Q_j = \sum_{k=0}^{\nu_j-1} \frac{N_j^kQ_j}{(z-\mu_j)^{k+1}}. \]

Multiplication by the coupling blocks gives the self energy formula for every \(z\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity

\[ \int_0^\infty {\mathrm e}^{-zt}{\mathrm e}^{\mu_jt} \frac{t^k}{k!} {\,\mathrm d} t = \frac{1}{(z-\mu_j)^{k+1}}. \]

Under stability,

\[ \int_0^\infty {\mathrm e}^{\operatorname{Re}\mu_jt} \frac{t^k}{k!} {\,\mathrm d} t = \frac{1}{(-\operatorname{Re}\mu_j)^{k+1}}. \]

Sum the omitted terms.

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Corollary E15.2: Optimal selection for the certified primary block bound

Under the stable assumptions of Theorem E15.1, define the primary block weights

\[ w_j = \sum_{k=0}^{\nu_j-1} \frac{{\left\lVert B_{j,k}\right\rVert}} {(-\operatorname{Re}\mu_j)^{k+1}}. \]

Among all sets \(J\) containing \(k\) primary blocks, the certified upper bound

\[ \sum_{j\notin J}w_j \]

is minimized by retaining the \(k\) largest weights. If

\[ w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)}, \]

then the least number of retained primary blocks certified to give \(L^1\) error at most \(\varepsilon\) is the least \(k\) for which

\[ \sum_{j>k}w_{(j)}\leq\varepsilon. \]

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Proof of Corollary E15.2

If a retained set contains a block with smaller weight than an omitted block, exchanging them reduces the omitted weight sum. Repeating the exchange gives the set of the \(k\) largest weights. The tolerance statement is the corresponding sorted tail condition.

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Corollary E15.3: Normal hidden dynamics

Suppose \(X_Q\) is a finite dimensional Hilbert space. If \(A_{QQ}\) is normal with spectral resolution

\[ A_{QQ}=\sum_{j=1}^r\mu_jQ_j, \]

then

\[ {\mathcal K}(t)=\sum_{j=1}^r{\mathrm e}^{\mu_jt}B_j, \qquad \Sigma_P(z)=\sum_{j=1}^r\frac{B_j}{z-\mu_j}, \]

where

\[ B_j=A_{PQ}Q_jA_{QP}. \]

If \(\operatorname{Re}\mu_j<0\), then

\[ {\left\lVert{\mathcal K}-{\mathcal K}_J\right\rVert}_{L^1(0,\infty)} \leq \sum_{j\notin J} \frac{{\left\lVert B_j\right\rVert}}{-\operatorname{Re}\mu_j}. \]

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Proof of Corollary E15.3

Normality removes every nilpotent part, so \(\nu_j=1\). Apply Theorem E15.1.

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Corollary E15.4: Pole order and hidden Jordan depth

Every finite hidden block produces a strictly proper rational self energy. At an eigenvalue \(\mu_j\), its pole order is at most the size of the largest Jordan block of \(A_{QQ}\) at \(\mu_j\). Equality holds when the corresponding highest coefficient \(B_{j,\nu_j-1}\) is nonzero.

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Proof of Corollary E15.4

Read the pole order from the partial fraction expansion in Theorem E15.1.

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A stable Jordan block produces a polynomial exponential memory mode. A diagonal hidden block produces only pure exponentials.

Corollary E15.5: Finite auxiliary state realization

Every kernel of the form

\[ {\mathcal K}(t)=B{\mathrm e}^{tD}C \]

can be represented exactly by adjoining the finite auxiliary state

\[ \dot y=Dy+Cx_P, \qquad \dot x_P=A_{PP}x_P+By, \qquad y(0)=0. \]

A nonzero auxiliary initial state adds the explicit forcing \(B{\mathrm e}^{tD}y(0)\). The self energy of the augmented system is

\[ B(zI-D)^{-1}C. \]

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Proof of Corollary E15.5

Eliminate \(y\) by variation of constants or apply Theorem E14.1 to the augmented block matrix.

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Theorem E15.6: Volterra stability under kernel compression

Let \(S(t)\) be a strongly continuous semigroup on a Banach space \(X\), fix \(T>0\), and assume

\[ {\left\lVert S(t)\right\rVert}\leq M_T \quad(0\leq t\leq T). \]

Let \(K,\widetilde K\in L^1((0,T),\mathcal B(X))\), let \(g\in L^1((0,T),X)\), and let \(x,\widetilde x\in C([0,T],X)\). Suppose these are mild solutions with the same initial state and forcing, except that \(K\) is replaced by \(\widetilde K\):

\[ \begin{aligned} x(t) ={}&S(t)x_0+ \int_0^tS(t-s)g(s){\,\mathrm d} s\\ &+ \int_0^tS(t-s) \int_0^sK(s-r)x(r){\,\mathrm d} r{\,\mathrm d} s. \end{aligned} \]

Assume

\[ \sup_{0\leq t\leq T}{\left\lVert\widetilde x(t)\right\rVert}\leq B. \]

Set

\[ L={\left\lVert K\right\rVert}_{L^1(0,T)}, \qquad \varepsilon={\left\lVert K-\widetilde K\right\rVert}_{L^1(0,T)}. \]

Then

\[ \sup_{0\leq t\leq T} {\left\lVert x(t)-\widetilde x(t)\right\rVert} \leq M_TBT\varepsilon \exp(M_TLT). \]

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Proof of Theorem E15.6

Let

\[ E(t)=\sup_{0\leq r\leq t}{\left\lVert x(r)-\widetilde x(r)\right\rVert}. \]

Subtract the two mild equations. Fubini's theorem and the triangle inequality give

\[ E(t) \leq M_TL\int_0^tE(r){\,\mathrm d} r + M_TBt\varepsilon. \]

For \(t\leq T\), replace the last factor \(t\) by \(T\) and apply Gronwall's inequality.

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Corollary E15.7: Trajectory bound for modal truncation

Under the assumptions of Theorem E15.1 and Theorem E15.6, replacing \({\mathcal K}\) by \({\mathcal K}_J\) changes the retained trajectory by at most

\[ M_TBT \exp(M_TLT) \sum_{j\notin J} \sum_{k=0}^{\nu_j-1} \frac{{\left\lVert B_{j,k}\right\rVert}} {(-\operatorname{Re}\mu_j)^{k+1}}. \]

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Proof of Corollary E15.7

Insert the \(L^1\) error from Theorem E15.1 into Theorem E15.6.

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Full and truncated memory kernels with their pointwise difference. The bound uses integrated error.

Remark E15.8

A finite hidden realization is exact for rational self energy. A general memory kernel may require infinitely many modes or a continuum of modes. The compression theorem gives a stability estimate once an \(L^1\) approximation has been supplied. The largest weight rule is optimal for the certified blockwise bound. It need not minimize the true kernel error when different modal terms cancel.

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