# Topology-Aware Exploration of Energy-Based Models Equilibrium: Toric QC-LDPC Codes and Hyperbolic MET QC-LDPC Codes

arXiv: https://arxiv.org/abs/2401.14749 · HTML: https://mlsanity.com/paper/2401.14749/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 70 references that are in the map.
- First posted on arXiv 26 Jan 2024.

## Builds on (most cited first, 13 of 13)

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- [Involution: Inverting the Inherence of Convolution for Visual Recognition](https://mlsanity.com/paper/2103.06255/) (2021)
- [Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models](https://mlsanity.com/paper/1605.07252/) (2016)
- [Topological Approaches to Deep Learning](https://mlsanity.com/paper/1811.01122/) (2018)
- [Six Lectures on Linearized Neural Networks](https://mlsanity.com/paper/2308.13431/) (2023)
- [Classification of Long Sequential Data using Circular Dilated Convolutional Neural Networks](https://mlsanity.com/paper/2201.02143/) (2022)
- [ChordMixer: A Scalable Neural Attention Model for Sequences with Different Lengths](https://mlsanity.com/paper/2206.05852/) (2022)
- [Codes on graphs: Models for elementary algebraic topology and statistical physics](https://mlsanity.com/paper/1707.06621/) (2017)
- [Boltzmann machines and energy-based models](https://mlsanity.com/paper/1708.06008/) (2017)
- [FAID Diversity via Neural Networks](https://mlsanity.com/paper/2105.04118/) (2021)
- [Belief Propagation in Conditional RBMs for Structured Prediction](https://mlsanity.com/paper/1703.00986/) (2017)

## Abstract (arXiv)

This paper presents a method for achieving equilibrium in the ISING Hamiltonian when confronted with unevenly distributed charges on an irregular grid. Employing (Multi-Edge) QC-LDPC codes and the Boltzmann machine, our approach involves dimensionally expanding the system, substituting charges with circulants, and representing distances through circulant shifts. This results in a systematic mapping of the charge system onto a space, transforming the irregular grid into a uniform configuration, applicable to Torical and Circular Hyperboloid Topologies. The paper covers fundamental definitions and notations related to QC-LDPC Codes, Multi-Edge QC-LDPC codes, and the Boltzmann machine. It explores the marginalization problem in code on the graph probabilistic models for evaluating the partition function, encompassing exact and approximate estimation techniques. Rigorous proof is provided for the attainability of equilibrium states for the Boltzmann machine under Torical and Circular Hyperboloid, paving the way for the application of our methodology. Practical applications of our approach are investigated in Finite Geometry QC-LDPC Codes, specifically in Material Science. The paper further explores its effectiveness in the realm of Natural Language Processing Transformer Deep Neural Networks, examining Generalized Repeat Accumulate Codes, Spatially-Coupled and Cage-Graph QC-LDPC Codes. The versatile and impactful nature of our topology-aware hardware-efficient quasi-cycle codes equilibrium method is showcased across diverse scientific domains without the use of specific section delineations.