Fitting an ellipsoid to random points: predictions using the replica method

TL;DR

This paper analyzes the problem of fitting an ellipsoid to high-dimensional Gaussian points, predicting a phase transition at a specific ratio using the replica method, and explores algorithmic solutions within this framework.

Contribution

It provides a non-rigorous replica method prediction of the SAT/UNSAT transition and characterizes the typical ellipsoid shape in the SAT phase, extending understanding of high-dimensional ellipsoid fitting.

Findings

Predicts a SAT/UNSAT transition at α=1/4

Characterizes the shape of typical fitting ellipsoids in the SAT phase

Shows nuclear norm minimization solves in the entire SAT phase

Abstract

We consider the problem of fitting a centered ellipsoid to $n$ standard Gaussian random vectors in $\mathbb{R}^d$, as $n, d \to \infty$ with $n/d^2 \to \alpha > 0$. It has been conjectured that this problem is, with high probability, satisfiable (SAT; that is, there exists an ellipsoid passing through all $n$ points) for $\alpha < 1/4$, and unsatisfiable (UNSAT) for $\alpha > 1/4$. In this work we give a precise analytical argument, based on the non-rigorous replica method of statistical physics, that indeed predicts a SAT/UNSAT transition at $\alpha = 1/4$, as well as the shape of a typical fitting ellipsoid in the SAT phase (i.e., the lengths of its principal axes). Besides the replica method, our main tool is the dilute limit of extensive-rank "HCIZ integrals" of random matrix theory. We further study different explicit algorithmic constructions of the matrix characterizing the ellipsoid. In particular, we show that a procedure based on minimizing its nuclear norm yields a solution in the whole SAT phase. Finally, we characterize the SAT/UNSAT transition for ellipsoid fitting of a large class of rotationally-invariant random vectors. Our work suggests mathematically rigorous ways to analyze fitting ellipsoids to random vectors, which is the topic of a companion work.