# Inverse problems for structured datasets using parallel TAP equations and RBM

## TL;DR

This paper introduces a parallel TAP-based algorithm for solving inverse problems in binary clustered datasets, demonstrating its effectiveness in the Hopfield model and exploring its relation to RBMs, with advantages over traditional methods.

## Contribution

It presents a novel parallel TAP equations approach for inverse problems, capable of retrieving exact patterns and handling large systems, and compares it with RBM methods.

## Key findings

- Parallel TAP equations accurately retrieve teacher patterns.
- AMP equations do not match expected behavior in the direct problem.
- The approach scales well to large system sizes.

## Abstract

We propose an efficient algorithm to solve inverse problems in the presence of binary clustered datasets. We consider the paradigmatic Hopfield model in a teacher student scenario, where this situation is found in the retrieval phase. This problem has been widely analyzed through various methods such as mean-field approaches or the pseudo-likelihood optimization. Our approach is based on the estimation of the posterior using the Thouless-Anderson-Palmer (TAP) equations in a parallel updating scheme. At the difference with other methods, it allows to retrieve the exact patterns of the teacher and the parallel update makes it possible to apply it for large system sizes. We also observe that the Approximate Message Passing (AMP) equations do not reproduce the expected behavior in the direct problem, questioning the standard practice used to obtain time indexes coming from Belief Propagation (BP). We tackle the same problem using a Restricted Boltzmann Machine (RBM) and discuss the analogies between the two algorithms.