Ellipsoid fitting up to constant via empirical covariance estimation
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The ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo and Willsky considers the maximum number n random Gaussian points in ℝ^d, such that with high probability, there exists an origin-symmetric ellipsoid passing through all the points. They conjectured a threshold of n = (1-o_d(1)) · d^2/4, while until recently, known lower bounds on the maximum possible n were of the form d^2/(log d)^O(1). We give a simple proof based on concentration of sample covariance matrices, that with probability 1 - o_d(1), it is possible to fit an ellipsoid through d^2/C random Gaussian points. Similar results were also obtained in two recent independent works by Hsieh, Kothari, Potechin and Xu [arXiv, July 2023] and by Bandeira, Maillard, Mendelson, and Paquette [arXiv, July 2023].
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