Computing Lewis Weights to High Precision

10/29/2021

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by   Maryam Fazel, et al.

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We present an algorithm for computing approximate ℓ_p Lewis weights to high precision. Given a full-rank 𝐀∈ℝ^m × n with m ≥ n and a scalar p>2, our algorithm computes ϵ-approximate ℓ_p Lewis weights of 𝐀 in O_p(log(1/ϵ)) 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∈ℝ^m × m. Prior to our work, such a computational complexity was known only for p ∈ (0, 4) [CohenPeng2015], and combined with this result, our work yields the first polylogarithmic-depth polynomial-work algorithm for the problem of computing ℓ_p 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.