The star-shaped space of solutions of the spherical negative perceptron

TL;DR

This paper analyzes the solution landscape of the spherical negative perceptron, revealing a star-shaped structure with a large connected component and a transition to disconnected solutions at higher constraint densities.

Contribution

It introduces a new analytical method for computing energy barriers and characterizes the star-shaped geometry of solutions in the spherical negative perceptron model.

Findings

Existence of a large geodesically convex component in the solution space.

Identification of a star-shaped subset of high-margin solutions.

A transition point where simple connectivity breaks down at higher constraint densities.

Abstract

Empirical studies on the landscape of neural networks have shown that low-energy configurations are often found in complex connected structures, where zero-energy paths between pairs of distant solutions can be constructed. Here we consider the spherical negative perceptron, a prototypical non-convex neural network model framed as a continuous constraint satisfaction problem. We introduce a general analytical method for computing energy barriers in the simplex with vertex configurations sampled from the equilibrium. We find that in the over-parameterized regime the solution manifold displays simple connectivity properties. There exists a large geodesically convex component that is attractive for a wide range of optimization dynamics. Inside this region we identify a subset of atypical high-margin solutions that are geodesically connected with most other solutions, giving rise to a star-shaped geometry. We analytically characterize the organization of the connected space of solutions and show numerical evidence of a transition, at larger constraint densities, where the aforementioned simple geodesic connectivity breaks down.