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 $n = \Omega( \, d^2/\mathrm{polylog}(d) \,)$, 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 convenient decomposition of a certain non-standard random matrix and a careful analysis of its Neumann expansion via the theory of graph matrices.

翻译：给定维度d中独立的标准高斯点v_1, ..., v_n，对于哪些(n, d)值，存在高概率下同时通过所有点的原点对称椭球体？这一拟合椭球体到随机点的基本问题与低秩矩阵分解、独立成分分析和主成分分析相关联。基于强有力的数值证据，Saunderson、Parrilo和Willsky [《决策与控制会议论文集》，第6031-6036页，2013] 推测，随着点数n增加，椭球体拟合问题从可行转变为不可行，并在n ~ d^2/4处出现尖锐阈值。我们通过为某些n = Ω( d^2/polylog(d) )构建拟合椭球体，将这一猜想解决到对数因子范围内，改进了Ghosh等人 [《计算机科学基础研讨会论文集》，第954-965页，2020] 要求n = o(d^{3/2})的先前工作。我们的证明展示了Saunderson等人最小二乘构造的可行性，利用了对某个非标准随机矩阵的便捷分解以及通过图矩阵理论对其Neumann展开的仔细分析。