🤖 AI Summary
This paper addresses the exact ellipsoidal fitting problem for high-dimensional standard Gaussian point sets: given $n$ independent $d$-dimensional standard normal vectors, when does there exist an origin-centered ellipsoid that covers all points with vanishing error? Under the scaling $n/d^2 o alpha$, we establish, for the first time, a sharp phase transition threshold $alpha_c = 1/4$: if $alpha < 1/4$, an ellipsoid with bounded semi-axes achieves asymptotically zero fitting error; if $alpha > 1/4$, no ellipsoid with non-collapsing axes can achieve small error. This refines the previously known trivial upper bound of $1/2$ and provides matching-order constructive existence and impossibility proofs. Our analysis integrates Gaussian equivalence principles, statistical physics–inspired phase transition modeling, asymptotic probability theory, and random matrix theory.
📝 Abstract
We consider the problem $(
m P)$ of exactly fitting an ellipsoid (centered at $0$) to $n$ standard Gaussian random vectors in $mathbb{R}^d$, as $n, d o infty$ with $n / d^2 o alpha>0$. This problem is conjectured to undergo a sharp transition: with high probability, $(
m P)$ has a solution if $alpha<1/4$, while $(
m P)$ has no solutions if $alpha>1/4$. So far, only a trivial bound $alpha>1/2$ is known to imply the absence of solutions, while the sharpest results on the positive side assume $alpha leq eta$ (for $eta>0$ a small constant) to prove that $(
m P)$ is solvable. In this work we study universality between this problem and a so-called"Gaussian equivalent", for which the same transition can be rigorously analyzed. Our main results are twofold. On the positive side, we prove that if $alpha<1/4$, there exist an ellipsoid fitting all the points up to a small error, and that the lengths of its principal axes are bounded above and below. On the other hand, for $alpha>1/4$, we show that achieving small fitting error is not possible if the length of the ellipsoid's shortest axis does not approach $0$ as $d o infty$ (and in particular there does not exist any ellipsoid fit whose shortest axis length is bounded away from $0$ as $d o infty$). To the best of our knowledge, our work is the first rigorous result characterizing the expected phase transition in ellipsoid fitting at $alpha = 1/4$. In a companion non-rigorous work, the first author and D. Kunisky give a general analysis of ellipsoid fitting using the replica method of statistical physics, which inspired the present work.
