Diffusion-Guided Feature Selection via Nishimori Temperature: Noise-Based Spectral Embedding

We propose Kohn-Sham Spectral Embedding (KSSE), an energy-based model replacing the top-layer classifier of convolutional networks with a sparse-graph spectral embedding at the Nishimori temperature of an associated Random-Bond Ising Model the spectral detectability threshold where class structure becomes marginally distinguishable from disorder. Mapping pre-trained features onto quasi-cyclic low-density parity-check graphs, we construct a regularized Laplacian (Bethe-Hessian) as an effective Kohn-Sham Hamiltonian, yielding D independent spectral problems-one per feature channel-solvable in $O(N log N + k_{mode}^{2} N)$ time by FFT on circulant blocks (Pontryagin self-duality), with low-mode Rayleigh-Ritz refinement ($k_{mode}=5$). Physically, this is a k.p effective-mass reduction on a one-dimensional ring crystal: the circulant support is the perfect crystal, the data weights a slowly varying impurity potential, and the Nishimori crossing a Fermi level at the band edge. Star-domain surgery optimizes the graph: instead of eliminating all frustrated cycles impossible without destroying the codewords-edge shifts create certified convexity around codewords with bounded residual frustration, with multi-scale fractal certification (basins $D_{2}<1$ vs rough landscapes $D_{2}>3$). The theory includes a generalized Ihara-Bass identity with a sharp spectral threshold, a non-backtracking growth trichotomy with frustration as a gauge-invariant $Z_{2}$ flux, a trapping-set spectral test, exact channel separability with a cup-product obstruction, plus loop-series, convexity, surgery, and quasi-stationarity bounds. On ImageNet-1000 with frozen EfficientNet-B4 features (D=1792) under a transductive protocol, KSSE achieves 88.93% Top-1 accuracy with ~21.24M parameters-beating Swin-L (197M, 86.4-87.3%) and matching the lower end of ViT-H/14 (632M, 88.0-89.5%) with 10x and 30x fewer parameters.

Mutual information (MI)-inspired feature learning techniques are capable of generating low-dimensional embeddings that retain nonlinear dependence structures, but direct estimations of MI suffer from noisy probability distribution estimates in the low-data regime. The H-Score objective, computed from second-order statistics, provides a practical proxy metric for training feature extraction networks. We prove that H-Score is invariant to invertible transformations in the unrestricted functional setting, but becomes sensitive to input basis rotations under constrained approximation classes. Consequently, we study unitary preconditioning for H-Score networks and show that selecting an appropriate basis rotation reduces finite-width truncation error by concentrating predictive dependence into fewer dominant modes. We identify the fast Fourier transform (FFT) as an effective data-independent, low-cost preconditioner for approximately stationary processes, where spectral structure induces concentration of the cross-covariance singular value spectrum. We introduce training-free metrics based on spectral entropy and cumulative dependence energy to quantify basis suitability and predict downstream inference gains prior to network training. Experiments across eight multivariate datasets demonstrate that FFT preconditioning is particularly useful in resource-constrained regimes, achieving up to 50% normalized mean squared error (NMSE) reduction, while the proposed metrics correlate with observed performance gains and correctly identify cases where spectral preconditioning is detrimental.

We identify a Selection Plateau phenomenon in one-shot neural network pruning: all rank-monotone weight scorers converge to identical accuracy at fixed sparsity, independent of functional form. We propose the Sparsity-Information-Complexity Spectrum (SICS) hypothesis: a sparsity-dependent minimum feature complexity kappa(S) governs plateau escape, with kappa=0 sufficient at low sparsity (S<0.65), kappa=1 dominant at critical sparsity (S~0.7), and kappa=2 necessary at extreme sparsity (S>0.75). On ViT-Small/CIFAR-10, testing nine feature classes across four sparsities, smooth non-monotone features provide +6.6% escape at S=0.7, while only raw features with high-frequency wiggle escape at S=0.8 (+2.6%). A fake non-monotone scorer underperforms the gradient baseline, indicating the requirement is magnitude-independent non-monotonicity. A handcrafted Gaussian bump achieves only +0.006 escape vs. chaos-derived +0.046, indicating rank-alignment is necessary but insufficient. SICS provides a unifying explanation for the performance clustering of diverse pruning methods and suggests that future selection algorithms should adapt feature complexity to target sparsity.