Counting stationary points of the loss function in the simplest constrained least-square optimization

TL;DR

This paper applies the Kac-Rice method to analyze the statistical properties of the loss landscape in a constrained least-squares problem, providing insights into the distribution of stationary points.

Contribution

It introduces a novel application of the Kac-Rice technique to characterize the optimization landscape of a constrained least-squares problem on a sphere.

Findings

Characterizes the distribution of stationary points.

Provides analytical expressions for the number of solutions.

Enhances understanding of the landscape complexity.

Abstract

We use Kac-Rice method to analyze statistical features of an "optimization landscape" of the loss function in a random version of the Oblique Procrustes Problem, one of the simplest optimization problems of the least-square type on a sphere.