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

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 $ρ(B_γ)\leq 1+δ$. 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 $Ω(1/\sqrt{d_{\min}})$; 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.

Planar-SfM: Camera Pose Estimation via Homography Graph Embeddings

Structure from Motion (SfM) systems traditionally struggle with planar scenes, where standard epipolar geometry-based methods become degenerate. Rather than viewing planar surfaces as a limitation, we propose a unified framework that leverages them as a source of geometric constraints. Our key insight is that each planar surface visible across multiple views provides an independent estimate of relative camera poses through homography decomposition. By aggregating estimates from multiple planes or even from a single dominant plane we achieve robust pose recovery in scenarios where traditional methods fail. We introduce a novel graph-based approach that constructs a pose-graph from homography estimates and employs spectral embedding to identify and filter unreliable edges. Our method maps homography-based pose estimates onto the real line based on their geometric and visual consistency, enabling efficient extraction of a maximally consistent spanning tree for pose recovery. This approach naturally handles both highly planar scenes, such as indoor sports arenas, and general $3$D environments. We demonstrate superior performance on basketball court imagery where existing methods struggle, while matching or exceeding state-of-the-art results on unconstrained outdoor scenes from the IMC Phototourism benchmark.