Faster Algorithm for Structured John Ellipsoid Computation

TL;DR

This paper introduces two faster algorithms for approximating the John Ellipsoid in convex, centrally symmetric polytopes, leveraging sketching and treewidth techniques to improve computational efficiency over previous methods.

Contribution

The paper presents novel algorithms that significantly reduce the running time for approximating the John Ellipsoid in specific convex polytopes, surpassing prior state-of-the-art methods.

Findings

Sketching-based algorithm runs in nearly input-sparsity time.

Treewidth-based algorithm runs in time proportional to the square of the treewidth.

Both algorithms outperform previous $\widetilde{O}(n d^2)$ methods.

Abstract

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].