Topology-Aware Sparse Factorization in Energy-Based Models: Tori QC-LDPC Codes and Circular Hyperboloid MET QC-LDPC Codes

This study proposed the intersection of Information Theory and Applied Topology, focusing on Codes on Graphs, particularly low-density parity-check (LDPC) codes. By exploiting the connection between the parity check matrix and Markov random fields, we symmetrically embed QC LDPC codes into the barycenters of energy minima within the Ising model. This embedding, enabled by toroidal and circular hyperboloid topologies of negative curvature, associates codewords with extrema and trapping sets with local minima. Strengthening LDPC codes with increased code distance enhances the separation of saddle points, improving the neural network's noise resistance (adversarial robustness) and representation power (capacity). The inherent block and sparse structure of toroidal and hyperboloid topolo-gies streamline KAM stability and preserve negative curvature in high-dimensional invariant quantities. This study aims to merge the KAM theorem, Non-equilibrium Torical Manifold, and Novikov conjectures within LDPC codes. Our constructive approach allows for synthesizing architectures of energy-efficient models tailored to codes on graphs with toric and hyperbolic topologies. We effectively represent Markov random fields in energy-efficient models using sparse block (quasi-cyclic) matrices (tensors) corresponding to hypergraphs (multigraphs), leading to a reduction in trainable parameters and a logarithmic decrease in tensor multiplexing complexity. Applying our approach to factorization problems on dense graphs, network issues, surface meshes, and covariance matrices demonstrates notable improve-ments in reconstruction accuracy using the Frobenius metric (up to 8 orders of magnitude in individual problems).