Algorithmic pure states for the negative spherical perceptron

10/29/2020
by   Ahmed El Alaoui, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/21/2023

Algorithmic Threshold for Multi-Species Spherical Spin Glasses

We study efficient optimization of the Hamiltonians of multi-species sph...
research
02/27/2023

Injectivity of ReLU networks: perspectives from statistical physics

When can the input of a ReLU neural network be inferred from its output?...
research
03/21/2018

The Hbot : A Holonomic Spherical Haptic Interface Driven by Non-Holonomic Wheels

We present the Hbot, a holonomic, singularity-free spherical robot desig...
research
12/23/2020

On spherical harmonics possessing octahedral symmetry

In this paper, we present the implicit representation of one special cla...
research
12/09/2022

Computability questions in the sphere packing problem

We consider the sets of dimensions for which there is an optimal sphere ...
research
08/30/2021

On the Number of Faces and Radii of Cells Induced by Gaussian Spherical Tessellations

We study a geometric property related to spherical hyperplane tessellati...
research
10/31/2019

Gaussian-Spherical Restricted Boltzmann Machines

We consider a special type of Restricted Boltzmann machine (RBM), namely...

Please sign up or login with your details

Forgot password? Click here to reset