Inverse problems in spin models

TL;DR

This paper develops methods to infer microscopic interactions in spin models like Ising and Hopfield models from collective behavior data, with explicit formulas and data requirements analysis.

Contribution

It provides explicit formulas for inferring couplings and patterns in spin models, and estimates data needed for accurate inference.

Findings

Explicit formulas for couplings in Ising models

Explicit formulas for pattern inference in Hopfield models

Analytical estimates of data requirements for inference

Abstract

Several recent experiments in biology study systems composed of several interacting elements, for example neuron networks. Normally, measurements describe only the collective behavior of the system, even if in most cases we would like to characterize how its different parts interact. The goal of this thesis is to extract information about the microscopic interactions as a function of their collective behavior for two different cases. First, we will study a system described by a generalized Ising model. We find explicit formulas for the couplings as a function of the correlations and magnetizations. In the following, we will study a system described by a Hopfield model. In this case, we find not only explicit formula for inferring the patterns, but also an analytical result that allows one to estimate how much data is necessary for a good inference.