# Low-rank combinatorial optimization and statistical learning by spatial photonic Ising machine

arXiv: https://arxiv.org/abs/2303.14993 · HTML: https://mlsanity.com/paper/2303.14993/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 48 references that are in the map.
- First posted on arXiv 27 Mar 2023.

## Builds on (most cited first, 5 of 5)

- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Application of QUBO solver using black-box optimization to structural design for resonance avoidance](https://mlsanity.com/paper/2204.04906/) (2022)
- [Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learning](https://mlsanity.com/paper/2112.11534/) (2021)
- [Sampling Approximately Low-Rank Ising Models: MCMC meets Variational Methods](https://mlsanity.com/paper/2202.08907/) (2022)
- [Efficient Solution of Boolean Satisfiability Problems with Digital MemComputing](https://mlsanity.com/paper/2011.06551/) (2020)

## Built on by (most cited first, 4 of 4)

- [Photonic restricted Boltzmann machine for content generation tasks](https://mlsanity.com/paper/2508.20472/) (2025)
- [Incorporating rank-free coupling and external field via an incoherent modulated spatial photonic Ising machine](https://mlsanity.com/paper/2512.21587/) (2025)
- [Optimizing Energy-based Neural Network Training with Coherent Ising Machine](https://mlsanity.com/paper/2606.09117/) (2026)
- [Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines](https://mlsanity.com/paper/2606.13454/) (2026)

## Abstract (arXiv)

The spatial photonic Ising machine (SPIM) [D. Pierangeli et al., Phys. Rev. Lett. 122, 213902 (2019)] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.