Kohn-Sham Spectral Embedding on Sparse Graphs at the Nishimori Temperature for Image Classification

Abstract

We propose Kohn-Sham Spectral Embedding (KSSE), an energy-based model replacing the dense classifier of convolutional neural networks with a sparse-graph spectral embedding evaluated at the Nishimori temperature of an associated Random-Bond Ising Model (RBIM). Mapping pre-trained features onto quasi-cyclic low-density parity-check graphs with a regularized Laplacian acting as a Kohn-Sham Hamiltonian decomposes the system into D independent single-channel spectral problems. These are solved in O(N log N + k_mode^2 N) time via the Fast Fourier Transform on circulant blocks a consequence of Pontryagin self-dualityâ with low-mode Rayleigh-Ritz refinement. Instead of eliminating all frustrated cycles, graph topology is optimized via star-domain surgery, using edge shifts to enforce certified local convexity around codewords while bounding residual frustration. Multi-scale fractal analysis and the learning-rate landscape certify the transition from rough landscapes to star-domain basins. Our rigorous theoretical framework establishes: a generalized Ihara-Bass identity linking belief propagation to the regularized Laplacian; a non-backtracking growth trichotomy where frustration enters as a gauge-invariant Z_2 flux; a trapping-set spectral test; an even-subgraph partition function expansion; exact additive separability with a cup-product obstruction; and a loop-series exchange-correlation bound certifying sub-percent factorization error at girth >= 6. Evaluated on ImageNet-1000 with frozen EfficientNet-B4 features under a transductive protocol, KSSE achieves 88.93% Top-1 accuracy using ~21.24M parameters, outperforming Swin-L (197M, 86.4-87.3%) and matching ViT-H/14 (632M, 88.0-89.5%) while reducing model size by 10x and 30x, respectively.