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- W4387322767 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." @default.
- W4387322767 created "2023-10-04" @default.
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- W4387322767 date "2023-10-02" @default.
- W4387322767 modified "2023-10-05" @default.
- W4387322767 title "Fitting an ellipsoid to random points: predictions using the replica method" @default.
- W4387322767 doi "https://doi.org/10.48550/arxiv.2310.01169" @default.
- W4387322767 hasPublicationYear "2023" @default.
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