Algorithmic Pure States for the Negative Spherical Perceptron

Abstract

We consider the spherical perceptron with Gaussian disorder. This is the set S of points \(\varvec{\sigma }\in \mathbb {R}^N\) on the sphere of radius \(\sqrt{N}\) satisfying \(\langle \varvec{g}_a , \varvec{\sigma }\rangle \ge \kappa \sqrt{N}\) for all \(1 \le a \le M\), where \((\varvec{g}_a)_{a=1}^M\) are independent standard gaussian vectors and \(\kappa \in \mathbb {R}\) is fixed. Various characteristics of S such as its measure and the largest M for which it is non-empty, were computed heuristically in statistical physics in the asymptotic regime \(N \rightarrow \infty \), \(M/N \rightarrow \alpha \). The case \(\kappa <0\) is of special interest as S is conjectured to exhibit a hierarchical tree-like geometry known as full replica-symmetry breaking (\(\text {FRSB}\)) close to the satisfiability threshold \(\alpha _{\text {SAT}}(\kappa )\), 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 \(\text {FRSB}\) structure for \(\kappa <0\) and outputs a vector \(\hat{\varvec{\sigma }}\) satisfying \(\langle \varvec{g}_a , \hat{\varvec{\sigma }}\rangle \ge \kappa \sqrt{N}\) 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 \(\hat{\varvec{\sigma }}\) to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that \(\bar{q}\rightarrow 1\) as \(\alpha \rightarrow \alpha _{\text {SAT}}(\kappa )\), so that \(\big \langle \varvec{g}_a, \hat{\varvec{\sigma }} \big \rangle / \vert \hat{\varvec{\sigma }} \vert \ge \kappa -o(1)\) near criticality.