Faster Algorithm for Structured John Ellipsoid Computation
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Computing John Ellipsoid is a fundamental problem in machine learning and convex optimization, where the goal is to compute the ellipsoid with maximal volume that lies in a given convex centrally symmetric polytope defined by a matrix A ∈ℝ^n × d. In this work, we show two faster algorithms for approximating the John Ellipsoid. ∙ For sparse matrix A, we can achieve nearly input sparsity time nnz(A) + d^ω, where ω is exponent of matrix multiplication. Currently, ω≈ 2.373. ∙ For the matrix A which has small treewidth τ, we can achieve n τ^2 time. Therefore, we significantly improves the state-of-the-art results on approximating the John Ellipsoid for centrally symmetric polytope [Cohen, Cousins, Lee, and Yang COLT 2019] which takes nd^2 time.
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