Faster Algorithms for Structured John Ellipsoid Computation

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].

Authors

• Yang Cao University of Science and Technology of China
• Xiaoyu Li Tencent
• Zhao Song Adobe Research
• Xin Yang Dalian University of Technology
• Tianyi Zhou

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