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This research proposes Kohn-Sham Spectral Embedding (KSSE), a physics-inspired architecture that replaces dense CNNs with sparse-graph spectral embeddings derived from Random-Bond Ising Models. By leveraging FFT on circulant blocks and optimizing graph topology via star-domain surgery, the method achieves competitive ImageNet accuracy while drastically reducing parameter count compared to standard Vision Transformers or Swin models. The work bridges condensed matter physics concepts like Nishimori temperature with practical deep learning efficiency for image classification tasks.
Kohn-Sham Spectral Embedding on Sparse Graphs at the Nishimori Temperature for Image Classification
arXiv:2607.28428v1 Announce Type: new 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 $\mathcal{O}(N\log N + k^2_{\text{mode}} N)$ time via FFT on circulant blocks (leveraging Pontryagin self-duality of $\mathbb{Z}/p\mathbb{Z}$) and low-order Rayleigh refinement. Graph topology is optimized using \emph{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 $\rho(B_\gamma)\leq 1+\delta$. Multi-scale fractal analysis ($D_2$ spectrum) and fractal learning-rate landscape certifies a landscape transition from rough regimes ($D_2>3$) to star-domain basins ($D_2<1$), enabling Rayleigh refinement with $k_{\text{mode}}=5$ 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 $\Omega(1/\sqrt{d_{\min}})$; 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 \textbf{88.93\%} Top-1 accuracy using $\approx 21.24$M 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 $10\times$ and $30\times$, respectively.
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 $\mathcal{O}(N\log N + k^2_{\text{mode}} N)$ time via FFT on circulant blocks (leveraging Pontryagin self-duality of $\mathbb{Z}/p\mathbb{Z}$) and low-order Rayleigh refinement. Graph topology is optimized using \emph{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 $\rho(B_\gamma)\leq 1+\delta$. Multi-scale fractal analysis ($D_2$ spectrum) and fractal learning-rate landscape certifies a landscape transition from rough regimes ($D_2>3$) to star-domain basins ($D_2<1$), enabling Rayleigh refinement with $k_{\text{mode}}=5$ 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 $\Omega(1/\sqrt{d_{\min}})$; 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 \textbf{88.93\%} Top-1 accuracy using $\approx 21.24$M 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 $10\times$ and $30\times$, respectively.
Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); Information Theory (cs.IT)
arXiv:2607.28428 [cs.LG]
(or arXiv:2607.28428v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.28428 arXiv-issued DOI via DataCite (pending registration)
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