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

Vasiliy Usatyuk, Denis Sapozhnikov, Sergey Egorov

Abstract

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.

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

Abstract

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.
Paper Structure (15 sections, 3 theorems, 68 equations, 31 figures)

This paper contains 15 sections, 3 theorems, 68 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Basic definition
  3. Multi-edge Type (Protograph) QC-LDPC Codes
  4. Spatially-Coupled Convolutional QC-LDPC Codes
  5. Boltzmann machine
  6. Marginalization Problem
  7. Topology-Aware Equilibrium
  8. Two-dimensional Ising Equilibrium Problem
  9. Interaction of Four Particles in Two-Dimensional Space Without Projecting Onto One Axis
  10. Interaction of Six Particles in Two-Dimensional Space Without Projecting Onto One Axis
  11. Circular Hyperboloid
  12. Example of Circular Hyperboloid: MET QC-LDPC Codes
  13. Implementing the Proposed Approach in Finite Geometry QC-LDPC Codes
  14. Demonstrating Equivalence: Generalized Repeat Accumulate Codes and Cage-Graph QC-LDPC Codes in NLP Transformer DNN
  15. CONCLUSION

Key Result

Lemma 1

Two particles with the same charge on a circle are in equilibrium if the left distance between the particles equals the right distance.

Figures (31)

  • Figure 1: (left) Tanner Graph of $H$ parity-check matrix (right) Multi-graph of $H_2$ parity-check matrix
  • Figure 2: Graphical representation of Trapping sets: a) TS(5,3), b) TS(4,4), Vasic09
  • Figure 3: Example of protograph and spatial coupled set of protograph
  • Figure 4: One $W \times C$ CPM-block of circulants $A(b_{(i,j)})$ shifted by values of vector D
  • Figure 5: Tail-biting Spatial-Couple QC-LDPC code parity-check matrix $C(H)$
  • ...and 26 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Definition 3
  • Definition 4