# Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

arXiv: https://arxiv.org/abs/2307.15778 · HTML: https://mlsanity.com/paper/2307.15778/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 191 references that are in the map.
- First posted on arXiv 28 Jul 2023.

## Builds on (most cited first, 20 of 20)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Long Range Arena: A Benchmark for Efficient Transformers](https://mlsanity.com/paper/2011.04006/) (2020)
- [Pay Less Attention with Lightweight and Dynamic Convolutions](https://mlsanity.com/paper/1901.10430/) (2019)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Why does deep and cheap learning work so well?](https://mlsanity.com/paper/1608.08225/) (2016)
- [Involution: Inverting the Inherence of Convolution for Visual Recognition](https://mlsanity.com/paper/2103.06255/) (2021)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Topological Approaches to Deep Learning](https://mlsanity.com/paper/1811.01122/) (2018)
- [Exposition and Interpretation of the Topology of Neural Networks](https://mlsanity.com/paper/1810.03234/) (2018)
- [Duality of Graphical Models and Tensor Networks](https://mlsanity.com/paper/1710.01437/) (2017)
- [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)
- [Q-FW: A Hybrid Classical-Quantum Frank-Wolfe for Quadratic Binary Optimization](https://mlsanity.com/paper/2203.12633/) (2022)
- [Nishimori meets Bethe: a spectral method for node classification in sparse weighted graphs](https://mlsanity.com/paper/2103.03561/) (2021)
- [Matrix factorization with neural networks](https://mlsanity.com/paper/2212.02105/) (2022)
- [Matrix factorization with Binary Components](https://mlsanity.com/paper/1401.6024/) (2014)
- [Codes on graphs: Models for elementary algebraic topology and statistical physics](https://mlsanity.com/paper/1707.06621/) (2017)
- [FAID Diversity via Neural Networks](https://mlsanity.com/paper/2105.04118/) (2021)
- [Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction](https://mlsanity.com/paper/2212.07201/) (2022)
- [Quantum Bridge Analytics II: Network Optimization and Combinatorial Chaining for Asset Exchange](https://mlsanity.com/paper/1911.03036/) (2019)

## Built on by (most cited first, 1 of 1)

- [Cyclic Group Projection for Enumerating Quasi-Cyclic Codes Trapping Sets](https://mlsanity.com/paper/2401.14810/) (2024)

## Abstract (arXiv)

The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.