The famous theorem of Fritz John states that any convex body has a unique maximal volume inscribed ellipsoid, known as the John Ellipsoid. Computing the John Ellipsoid is a fundamental problem in convex optimization. In this paper, we focus on approximating the John Ellipsoid inscribed in a convex and centrally symmetric polytope defined by $ P := \{ x \in \mathbb{R}^d : -\mathbf{1}_n \leq A x \leq \mathbf{1}_n \},$ where $ A \in \mathbb{R}^{n \times d} $ is a rank-$d$ matrix and $ \mathbf{1}_n \in \mathbb{R}^n $ is the all-ones vector. We develop two efficient algorithms for approximating the John Ellipsoid. The first is a sketching-based algorithm that runs in nearly input-sparsity time $ \widetilde{O}(\mathrm{nnz}(A) + d^\omega) $, where $ \mathrm{nnz}(A) $ denotes the number of nonzero entries in the matrix $A$ and $ \omega \approx 2.37$ is the current matrix multiplication exponent. The second is a treewidth-based algorithm that runs in time $ \widetilde{O}(n \tau^2)$, where $\tau$ is the treewidth of the dual graph of the matrix $A$. Our algorithms significantly improve upon the state-of-the-art running time of $ \widetilde{O}(n d^2) $ achieved by [Cohen, Cousins, Lee, and Yang, COLT 2019].

翻译：弗里茨·约翰的著名定理指出，任何凸体都存在唯一的最大体积内接椭球，称为约翰椭球。计算约翰椭球是凸优化中的一个基本问题。本文聚焦于近似计算内接于一个凸且中心对称多面体的约翰椭球，该多面体定义为 $ P := \{ x \in \mathbb{R}^d : -\mathbf{1}_n \leq A x \leq \mathbf{1}_n \},$ 其中 $ A \in \mathbb{R}^{n \times d} $ 是一个秩为 $d$ 的矩阵，$ \mathbf{1}_n \in \mathbb{R}^n $ 是全1向量。我们开发了两种近似计算约翰椭球的高效算法。第一种是基于草图技术的算法，其运行时间接近输入稀疏度 $ \widetilde{O}(\mathrm{nnz}(A) + d^\omega) $，其中 $ \mathrm{nnz}(A) $ 表示矩阵 $A$ 的非零元素数量，$ \omega \approx 2.37$ 是当前矩阵乘法指数。第二种是基于树宽的算法，其运行时间为 $ \widetilde{O}(n \tau^2)$，其中 $\tau$ 是矩阵 $A$ 对偶图的树宽。我们的算法显著改进了由 [Cohen, Cousins, Lee, and Yang, COLT 2019] 实现的最先进运行时间 $ \widetilde{O}(n d^2) $。