Gaussian–T³ Spectral Inference and Sparse-PCA Recovery

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A Gaussian heat channel on the three-torus has exact Fourier multipliers e−τ|m|2: all modes remain identifiable, but their exponentially small singular values make functional recovery severely ill posed. We give explicit Fourier range criteria for proximal bridge functions, minimax logarithmic Sobolev recovery rates, and conditions for stable minimal T3 predictive representations. These implications concern specified statistical experiments, not arbitrary data distributions. For spiked Gaussian covariance Ip + λvv⊤, we study the computational and statistical distinc- tion between general signed-sparse spikes and registered geometric subfamilies. The cumulative theorems include exact Gaussian likelihood and information identities, lower bounds for restricted low-rank/feedback measurement architectures, finite-dimensional rank and symmetry obstruc- tions, and several explicit polynomial-time receivers at their stated sample budgets. A lossless registered Gaussian heat–torus–solid-torus–M¨obius composition has an exact left inverse and preserves the underlying experiment, without creating additional observations. The central registered-orbit result treats the variable-length M¨obius-signed circular-interval translation dictionary. Its parameter-free maximum-likelihood ranking is computable in O(NpKmax) arithmetic and has exact recovery with failure at most δ under N = Oλ(k log(e/δ) + log(p/δ)) original full Gaussian samples, where k is the unknown true interval length. The improved additive bound uses an explicit circular boundary-shell count and a rank-one Gaussian Hellinger comparison. Matching information lower bounds hold for the specified orbit union. A further exact composition handles arbitrary relative signs and unknown sparsity on any polynomially enumerable … [source excerpt; see full PDF]

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