Counting solutions from finite samplings

TL;DR

This paper introduces a method using inverse Ising models and belief propagation to estimate the number of solutions in constraint satisfaction problems and neural data, showing accuracy in various models and real neural systems.

Contribution

It presents a novel approach to estimate solution counts via entropy approximation with belief propagation, applicable to both computational models and neural data.

Findings

Accurate solution count estimates for small constraint densities in SAT problems.

Effective entropy estimation for the binary perceptron model.

Consistent solution size predictions for neural network data.

Abstract

We formulate the solution counting problem within the framework of inverse Ising problem and use fast belief propagation equations to estimate the entropy whose value provides an estimate on the true one. We test this idea on both diluted models (random 2-SAT and 3-SAT problems) and fully-connected model (binary perceptron), and show that when the constraint density is small, this estimate can be very close to the true value. The information stored by the salamander retina under the natural movie stimuli can also be estimated and our result is consistent with that obtained by Monte Carlo method. Of particular significance is sizes of other metastable states for this real neuronal network are predicted.