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

2026/07/30 by V. S. Usatyuk, D. A. Sapozhnikov, S. I. Egorov
Computer Science · Mathematics · #cs.LG #cs.CV #cs.IT #math.IT

paper · pdf

42 pages, 10 figures, 5 tables, was presented at the 10th International Conference 'Deep Learning on Computational Physics (DLCP2026)', under review for the Moscow University Physics Bulletin, Physics series

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We introduce Kohn--Sham Spectral Embedding (KSSE), a physics-inspired energy-based model replacing dense CNN classifiers with a sparse-graph spectral embedding evaluated at the Nishimori temperature of an associated Random-Bond Ising Model. By mapping pre-trained features onto quasi-cyclic low-density parity-check graphs and constructing a regularized Laplacian acting as a Kohn--Sham Hamiltonian, we solve D independent channel spectral problems in O(Nlog N + k²_mode N) time via FFT on circulant blocks (leveraging Pontryagin self-duality of ℤ/pℤ) and low-order Rayleigh refinement. Graph topology is optimized using star-domain surgery: rather than destroying information-carrying codewords by removing frustrated cycles, we construct edge shifts creating local convexity around codewords while bounding residual frustration to ρ(B_γ)≤ 1+δ. Multi-scale fractal analysis (D₂ spectrum) and fractal learning-rate landscape certifies a landscape transition from rough regimes (D₂>3) to star-domain basins (D₂<1), enabling Rayleigh refinement with k_mode=5 modes. We prove six theoretical results: a generalized Ihara--Bass identity linking belief propagation to the Laplacian; trapping-set eigenvalue correspondence; additive channel separability with an explicit exchange-correlation bound; a surgery theorem bounding frustration with attractor width Ω(1/√d_min); a quasi-stationarity perturbation bound; and a fixed-point convergence theorem. In a transductive protocol on ImageNet-1000 with frozen EfficientNet-B4 features (D=1792), 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%) under standard inductive setups, while reducing model footprint by 10× and 30×, respectively.

Citations

• Natural Image Classification via Quasi-Cyclic Graph Ensembles and Random-Bond Ising Models at the Nishimori Temperature
• A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the N^1/3 distance barrier
• The boundary of neural network trainability is fractal
• Cyclic Group Projection for Enumerating Quasi-Cyclic Codes Trapping Sets
• An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale
• EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks
• On the Hardness of Approximating Stopping and Trapping Sets in LDPC Codes
• Measuring the strangeness of strange attractors
• Self-Consistent Equations Including Exchange and Correlation Effects
• Swin Transformer: Hierarchical Vision Transformer using Shifted Windows
• Swin Transformer: Hierarchical Vision Transformer using Shifted Windows

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