Algorithmic pure states for the negative spherical perceptron

Abstract

We consider the spherical perceptron with Gaussian disorder. This is the set $S$ of points $σ\in \mathbb{R}^N$ on the sphere of radius $\sqrt{N}$ satisfying $\langle g_a , σ\rangle \ge κ\sqrt{N}\,$ for all $1 \le a \le M$, where $(g_a)_{a=1}^M$ are independent standard gaussian vectors and $κ\in \mathbb{R}$ is fixed. Various characteristics of $S$ such as its surface measure and the largest $M$ for which it is non-empty, were computed heuristically in statistical physics in the asymptotic regime $N \to \infty$, $M/N \to α$. The case $κ<0$ is of special interest as $S$ is conjectured to exhibit a hierarchical tree-like geometry known as "full replica-symmetry breaking" (FRSB) close to the satisfiability threshold $α_{\text{SAT}}(κ)$, and whose characteristics are captured by a Parisi variational principle akin to the one appearing in the Sherrington-Kirkpatrick model. In this paper we design an efficient algorithm which, given oracle access to the solution of the Parisi variational principle, exploits this conjectured FRSB structure for $κ<0$ and outputs a vector $\hatσ$ satisfying $\langle g_a , \hatσ\rangle \ge κ\sqrt{N}$ for all $1\le a \le M$ and lying on a sphere of non-trivial radius $\sqrt{\bar{q} N}$, where $\bar{q} \in (0,1)$ is the right-end of the support of the associated Parisi measure. We expect $\hatσ$ to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that $\bar{q} \to 1$ as $α\to α_{\text{SAT}}(κ)$, so that $\big\langle g_a,\hatσ/|\hatσ|\big\rangle \geq (κ-o(1))\sqrt{N}$ near criticality.