Computing Lewis Weights to High Precision
Abstract
We present an algorithm for computing approximate $\ell_p$ Lewis weights to high precision. Given a fullrank $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $m \geq n$ and a scalar $p>2$, our algorithm computes $\epsilon$approximate $\ell_p$ Lewis weights of $\mathbf{A}$ in $\widetilde{O}_p(\log(1/\epsilon))$ iterations; the cost of each iteration is linear in the input size plus the cost of computing the leverage scores of $\mathbf{D}\mathbf{A}$ for diagonal $\mathbf{D} \in \mathbb{R}^{m \times m}$. Prior to our work, such a computational complexity was known only for $p \in (0, 4)$ [CohenPeng2015], and combined with this result, our work yields the first polylogarithmicdepth polynomialwork algorithm for the problem of computing $\ell_p$ Lewis weights to high precision for all constant $p > 0$. An important consequence of this result is also the first polylogarithmicdepth polynomialwork algorithm for computing a nearly optimal selfconcordant barrier for a polytope.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 DOI:
 10.48550/arXiv.2110.15563
 arXiv:
 arXiv:2110.15563
 Bibcode:
 2021arXiv211015563F
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Optimization and Control
 EPrint:
 24 pages