In [Sau11,SPW13], Saunderson, Parrilo and Willsky asked the following elegant geometric question: what is the largest $m= m(d)$ such that there is an ellipsoid in $\mathbb{R}^d$ that passes through $v_1, v_2, \ldots, v_m$ with high probability when the $v_i$s are chosen independently from the standard Gaussian distribution $N(0,I_{d})$. The existence of such an ellipsoid is equivalent to the existence of a positive semidefinite matrix $X$ such that $v_i^{\top}X v_i =1$ for every $1 \leq i \leq m$ - a natural example of a random semidefinite program. SPW conjectured that $m= (1-o(1)) d^2/4$ with high probability. Very recently, Potechin, Turner, Venkat and Wein and Kane and Diakonikolas proved that $m \geq d^2/\log^{O(1)}(d)$ via certain explicit constructions. In this work, we give a substantially tighter analysis of their construction to prove that $m \geq d^2/C$ for an absolute constant $C>0$. This resolves one direction of the SPW conjecture up to a constant. Our analysis proceeds via the method of Graphical Matrix Decomposition that has recently been used to analyze correlated random matrices arising in various areas [BHK+19]. Our key new technical tool is a refined method to prove singular value upper bounds on certain correlated random matrices that are tight up to absolute dimension-independent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension.

翻译：在[Sau11, SPW13]中，Saunderson、Parrilo和Willsky提出了以下优雅的几何问题：当$v_i$独立地服从标准高斯分布$N(0,I_d)$时，能使$\mathbb{R}^d$中存在一个椭球以高概率经过$v_1, v_2, \ldots, v_m$的最大$m=m(d)$是多少？该椭球的存在等价于存在一个半正定矩阵$X$，使得对所有$1 \leq i \leq m$有$v_i^{\top}X v_i = 1$——这是一个随机半定规划的自然实例。SPW猜想$m= (1-o(1)) d^2/4$以高概率成立。最近，Potechin、Turner、Venkat与Wein以及Kane与Diakonikolas通过某些显式构造证明了$m \geq d^2/\log^{O(1)}(d)$。本文中，我们对其构造进行了更紧致的分析，证明对于绝对常数$C>0$，有$m \geq d^2/C$。这将在常数精度下解决SPW猜想的一个方向。我们的分析采用了图形矩阵分解方法，该方法近期已被用于分析不同领域中出现的相关随机矩阵[BHK+19]。我们关键的新技术工具是证明某些相关随机矩阵奇异值上界的一种精细化方法，该上界在绝对维数无关常数意义下是紧致的。相比之下，以往分析此类矩阵的所有方法都会损失维度的对数因子。