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

Antoine Maillard, Dmitriy Kunisky

TL;DR

This work analyzes the problem of fitting a centered ellipsoid to $n$ standard Gaussian vectors in high dimension, reframing it as a semidefinite program and exploring the SAT/UNSAT transition in the regime $n/d^2\to\alpha$. Employing the non-rigorous replica method together with the dilute limit of extensive-rank HCIZ integrals, the authors predict a sharp SAT/UNSAT threshold at $\alpha_c=1/4$, characterize the typical ellipsoid shape in the SAT phase, and derive the minimal axis lengths. They also study the performance of explicit estimators, notably the minimal nuclear-norm solution, which remains PSD throughout the SAT phase, and extend the analysis to rotationally invariant vectors with norm fluctuations parametrized by $\tau$, obtaining $\alpha_c(\tau)$. The results connect to Gaussian-equivalent problems and offer mathematically guided routes toward rigorous proofs, with a companion work providing rigorous validation for a modified problem. The combination of replica theory and HCIZ techniques yields a detailed, quantitative picture of the solution space geometry and algorithmic implications for ellipsoid fitting in high dimensions.

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 $n/d^2 \to α> 0$. 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 $α< 1/4$, and unsatisfiable (UNSAT) for $α> 1/4$. 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 $α= 1/4$, 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.

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

TL;DR

This work analyzes the problem of fitting a centered ellipsoid to standard Gaussian vectors in high dimension, reframing it as a semidefinite program and exploring the SAT/UNSAT transition in the regime . Employing the non-rigorous replica method together with the dilute limit of extensive-rank HCIZ integrals, the authors predict a sharp SAT/UNSAT threshold at , characterize the typical ellipsoid shape in the SAT phase, and derive the minimal axis lengths. They also study the performance of explicit estimators, notably the minimal nuclear-norm solution, which remains PSD throughout the SAT phase, and extend the analysis to rotationally invariant vectors with norm fluctuations parametrized by , obtaining . The results connect to Gaussian-equivalent problems and offer mathematically guided routes toward rigorous proofs, with a companion work providing rigorous validation for a modified problem. The combination of replica theory and HCIZ techniques yields a detailed, quantitative picture of the solution space geometry and algorithmic implications for ellipsoid fitting in high dimensions.

Abstract

We consider the problem of fitting a centered ellipsoid to standard Gaussian random vectors in , as with . It has been conjectured that this problem is, with high probability, satisfiable (SAT; that is, there exists an ellipsoid passing through all points) for , and unsatisfiable (UNSAT) for . 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 , 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.
Paper Structure (38 sections, 1 theorem, 121 equations, 10 figures)

This paper contains 38 sections, 1 theorem, 121 equations, 10 figures.

Table of Contents

  1. Introduction
  2. General remarks
  3. Related literature
  4. Ellipsoid fitting
  5. Statistical physics approaches
  6. Summary of main results
  7. The SAT/UNSAT transition and solutions in the SAT phase
  8. The thresholds of different explicit approaches
  9. Beyond Gaussian vectors: general norm fluctuations
  10. Organization of the paper
  11. Replica-symmetric analysis of the entropy of solutions
  12. The free entropy
  13. Replicated free entropy
  14. Final result
  15. The dilute limit of HCIZ integrals and consequences
  16. ...and 23 more sections

Key Result

Theorem 1

Let $d \geq 1$, and ${\textbf{A}},{\textbf{B}} \in \mathcal{S}_d$. We assume that the empirical spectral distributions of ${\textbf{A}}$ and ${\textbf{B}}$ both converge as $d \to \infty$ to probability measures $\rho_A, \rho_B \in \mathcal{M}_1^+(\mathbb{R})$. Then: Here we have defined: Moreover, $f(t, x) \coloneqq v_t(x) + i \pi \rho_t(x)$ satisfies a complex Burgers' equation with prescribed

Figures (10)

  • Figure 1: A summary of the current state of the ellipsoid fitting conjecture. In red, we show regions which are rigorously known to be in the UNSAT phase, and in orange regions which are conjectured to be. Similarly, we show in green regions rigorously known to be in the SAT phase, and in yellow regions which are conjectured to be so. The companion work to this manuscript maillard2023fitting closes these gaps by proving rigorously that a slightly modified problem has a transition at $n \simeq d^2/4$.
  • Figure 2: Evolution of the solution set and typical spectral density of solutions (left), and the typical spectral density near the transition (right). For any $\alpha < 1/4$, the spectral density and the volume of the solution space are characterized by the "replica" equations \ref{['eq:phi_RS_general']},\ref{['eq:se_general']}.
  • Figure 3: The evolution of $\alpha_c(\kappa)$ (left) and of $\ell^\star(\alpha) = [\kappa^\star(\alpha)]^{-1/2}$, the minimal value of the longest principal axis of any ellipsoid fit (right).
  • Figure 4: The thresholds for different explicit constructions of a solution to the linear constraints. For any given method with threshold $\alpha_c$, the solution ceases to be positive semidefinite for $n> \alpha_c d^2$. Notice that the minimal nuclear norm approach succeeds in the whole SAT phase. For each method, we analytically derive $\alpha_c$, and we furthermore predict the asymptotic spectral density of solutions, see Claim \ref{['claim:explicit_constructions']}.
  • Figure 5: The SAT/UNSAT transition for rotationally-invariant vectors with fluctuating norm.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Claim 1: Solution space of ellipsoid fitting
  • Claim 2: Performance of explicit constructions
  • Claim 3: SAT/UNSAT transition for general rotationally-invariant distributions
  • Theorem 1: Extensive-rank HCIZ integral -- informal matytsin1994largeguionnet2002large