Algorithmic pure states for the negative spherical perceptron
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We consider the spherical perceptron with Gaussian disorder. This is the set S of points σ∈ℝ^N on the sphere of radius √(N) satisfying ⟨ g_a , σ⟩≥κ√(N) for all 1 ≤ a ≤ M, where (g_a)_a=1^M are independent standard gaussian vectors and κ∈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 →∞, M/N →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_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 σ̂ satisfying ⟨ g_a , σ̂⟩≥κ√(N) for all 1≤ a ≤ M and lying on a sphere of non-trivial radius √(q̅ N), where q̅∈ (0,1) is the right-end of the support of the associated Parisi measure. We expect σ̂ to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that q̅→ 1 as α→α_SAT(κ), so that ⟨ g_a,σ̂/|σ̂|⟩≥ (κ-o(1))√(N) near criticality.
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