# Programmable k-local Ising interactions and shallow optical Kolmogorov--Arnold networks through repeated data encounters

arXiv: https://arxiv.org/abs/2508.17440 · HTML: https://mlsanity.com/paper/2508.17440/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 45 references that are in the map.
- First posted on arXiv 24 Aug 2025.

## Builds on (most cited first, 7 of 7)

- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Nonlinear Processing with Linear Optics](https://mlsanity.com/paper/2307.08533/) (2023)
- [Efficient Optimization with Higher-Order Ising Machines](https://mlsanity.com/paper/2212.03426/) (2022)
- [All-to-all reconfigurability with sparse and higher-order Ising machines](https://mlsanity.com/paper/2312.08748/) (2023)
- [Analog Iterative Machine (AIM): using light to solve quadratic optimization problems with mixed variables](https://mlsanity.com/paper/2304.12594/) (2023)
- [Exact Spin Elimination for Quadratic and k-Local Ising Optimization](https://mlsanity.com/paper/2505.07163/) (2025)
- [Higher-Order Neuromorphic Ising Machines -- Autoencoders and Fowler-Nordheim Annealers are all you need for Scalability](https://mlsanity.com/paper/2506.19964/) (2025)

## Built on by (most cited first, 2 of 2)

- [Learning Nonlinear Heterogeneity in Physical Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2601.15340/) (2026)
- [KANEL: Kolmogorov-Arnold Network Ensemble Learning Enables Early Hit Enrichment in High-Throughput Virtual Screening](https://mlsanity.com/paper/2603.25755/) (2026)

## Abstract (arXiv)

Photonic processors are naturally suited to linear transformations, but independently programmable higher-order interactions usually require nonlinear media or a reduction to pairwise models. We introduce a repeated-encounter architecture that combines linear optical propagation with square-law detection to evaluate sparse and structured k-local Ising objectives. Each hyperedge is routed to a resolved channel, returned through the spin-dependent mask, and reconstructed from calibrated encounter-order signals. Within the stated architecture class, a k-spin Walsh term requires at least lceil k/2rceil data encounters. A reciprocal rank-one recollection attains this bound for every finite order, allowing hyperedge identity, interaction order, and signed coupling to be programmed independently without quadratization ancillas or material optical nonlinearities. We test the R=2, k=4 member in a finite discrete-Fourier model of an ideal folded 4f relay. Reciprocal recollection produces the four-body response, whereas a fixed-patch control does not; configuration-level calibration measures the leakage caused by finite windows. Under signed-amplitude encoding, the same encounter hierarchy spans polynomial edge functions for shallow optical Kolmogorov--Arnold networks. The arbitrary-k result is analytic; the finite Fourier calculation tests its two-encounter member and does not replace experimental validation.