2. Exact realization and atlas consistency
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2.1 A left inverse is the reconstruction contract
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Let \(X\) and \(C_a\) be finite-dimensional Hilbert spaces. A linear atlas consists of analysis and synthesis maps
The coefficients representing actual signals form \(\mathcal C_a=\operatorname{im}A_a\). This distinction matters for redundant representations, where \(\mathcal C_a\) is a proper subspace of the ambient coefficient space.
Proposition 1: Exact realization and consistency
The analysis map is injective, \(\Pi_a=A_aS_a\) is an idempotent with image \(\mathcal C_a\), and \(\Pi_ac=c\) exactly when \(c\) is a realizable coefficient vector. Moreover,
The first inequality is understood for nonzero \(X\). No assertion of orthogonality follows from a left inverse alone.
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Proof
If \(A_ax=0\), then \(x=S_aA_ax=0\). Also \(\Pi_a^2=A_a(S_aA_a)S_a=\Pi_a\). Its image is contained in \(\mathcal C_a\) and it fixes every \(A_ax\), so its image is exactly \(\mathcal C_a\). Both norm inequalities follow from the operator-norm inequality applied to \(S_a\). \(\square\)
For a unitary atlas \(U_a\), the synthesis is \(U_a^*\) and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.
2.2 Several exact atlases do not create new observations
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For unitary \(U_a\) and positive weights \(w_a\), put
Then \(\mathcal S\mathcal A=I\) and \(\|\mathcal Ax\|^2=c\|x\|^2\), as proved in [K1]. Here \(\mathcal A\mathcal S=c^{-1}\mathcal A\mathcal A^*\) is the orthogonal projection onto the consistent coefficient subspace.
Each complete atlas contains the same observed information because it has a left inverse. Multiple atlases offer different sparsity, locality, conditioning and operator descriptions. They do not provide independent observations of the world. In particular, multiplying likelihoods from deterministic recodings as though they were independent measurements generally counts the same evidence more than once.
Categorical one-hot encoding is a separately typed injective map of sets. It is not a unitary map on the scalar-valued signal space. Its exact decoder is valid on one-hot arrays, not on arbitrary coefficient vectors. The linear tight-frame theorem does not apply to it without a separately specified linear embedding and inner product. This separation is already explicit in the capstone's categorical section.