On the topology of solutions to random continuous constraint satisfaction problems

TL;DR

This paper analyzes the topology of solution manifolds for random polynomial equations on spheres, revealing phase transitions in their Euler characteristic related to problem parameters, with implications for understanding energy landscapes in spin glasses.

Contribution

It introduces a large-N analysis of the Euler characteristic of solution manifolds in random polynomial CSPs, identifying phase transitions and conjecturing links to spin glass energy dynamics.

Findings

Identification of five topological phases based on parameters

Explicit computation of average Euler characteristic in large N limit

Conjectured connection between topological transition and gradient descent energy

Abstract

We consider the set of solutions to $M$ random polynomial equations whose $N$ variables are restricted to the $(N-1)$-sphere. Each equation has independent Gaussian coefficients and a target value $V_0$. When solutions exist, they form a manifold. We compute the average Euler characteristic of this manifold in the limit of large $N$, and find different behavior depending on the target value $V_0$, the ratio $\alpha=M/N$, and the variances of the coefficients. We divide this behavior into five phases with different implications for the topology of the solution manifold. When $M=1$ there is a correspondence between this problem and level sets of the energy in the spherical spin glasses. We conjecture that the transition energy dividing two of the topological phases corresponds to the energy asymptotically reached by gradient descent from a random initial condition, possibly resolving an open problem in out-of-equilibrium dynamics. However, the quality of the available data leaves the question open for now.