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 Rd: 1n Ax 1n}, where A Rn d is a rank-d matrix and 1n Rn is the all-ones vector. We develop two efficient algorithms for approximating the John Ellipsoid. The first is a sketchingbased algorithm that runs in nearly input-sparsity time eO(nnz(A)+dω), where nnz(A)denotes the number of nonzero entries in the matrix Aand ω 2.37is the current matrix multiplication exponent. The second is a treewidth-based algorithm that runs in time eO(nτ2), where τ is the treewidth of the dual graph of the matrix A. Our algorithms significantly improve upon the state-of-the-art running time of eO(nd2)achieved by [Cohen, Cousins, Lee, and Yang, COLT 2019].