Given independent standard Gaussian points $v_1, \ldots, v_n$ in dimension $d$, for what values of $(n, d)$ does there exist with high probability an origin-symmetric ellipsoid that simultaneously passes through all of the points? This basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. Based on strong numerical evidence, Saunderson, Parrilo, and Willsky [Proc. of Conference on Decision and Control, pp. 6031-6036, 2013] conjecture that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points $n$ increases, with a sharp threshold at $n \sim d^2/4$. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = \Omega( \, d^2/\log^5(d) \,)$, improving prior work of Ghosh et al. [Proc. of Symposium on Foundations of Computer Science, pp. 954-965, 2020] that requires $n = o(d^{3/2})$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a careful analysis of the eigenvectors and eigenvalues of a certain non-standard random matrix.
翻译:根据独立的标准高斯点 $_1,\ldots, v_n美元(维值为美元),对于美元值(n, d)是否存在极有可能同时通过所有点的源对数闪光度?将闪光度调整到随机点这一基本问题与低位矩阵分解、独立部件分析和主要组成部分分析有关。根据强有力的数字证据,Saundson, Parrilo和Willsky[决定和控制会议进程, pp. 6031-606, 2013] 推测,随机安装问题从可行转向不可行,因为点数增加为美元,而临界值为美元=%2/4美元。我们通过为一些美元=Omega(\, d2/log5)和Willsky[决定和控制会议进程,pp. 603, 2013] 随机安装问题从可行过渡到不可行,因为点数增加,门槛为$n d’2/4美元。我们通过为某些基数=Omegas, dmexmission*5\\\\\, legrodual lexal exuratealmacials exisals excialus.