Computing Lewis Weights to High Precision

Maryam Fazel, Yin Tat Lee, Swati Padmanabhan, Aaron Sidford

Abstract: We present an algorithm for computing approximate ℓp Lewis weights to high precision. Given a full-rank A ∊ ℝm × n with m ≥ n and a scalar p > 2, our algorithm computes ∊-approximate ℓp Lewis weights of A in Õp(log(1/∊)) iterations; the cost of each iteration is linear in the input size plus the cost of computing the leverage scores of DA for diagonal D ∊ ℝm × m. Prior to our work, such a computational complexity was known only for p ∊ (0,4) [CP15], and combined with this result, our work yields the first polylogarithmic-depth polynomial-work algorithm for the problem of computing ℓp Lewis weights to high precision for all constant p > 0. An important consequence of this result is also the first polylogarithmic-depth polynomial-work algorithm for computing a nearly optimal self-concordant barrier for a polytope.