Regularized Weighted Low Rank Approximation

TL;DR

This paper introduces a new approach to regularized weighted low rank approximation, providing sharper theoretical guarantees and faster algorithms by leveraging advanced matrix concentration bounds and structural insights.

Contribution

It develops parameterized complexity bounds based on statistical dimension, enabling rank-independent and more efficient algorithms for a challenging NP-hard problem.

Findings

Sharper guarantees for regularized weighted low rank approximation.

Parameterization based on statistical dimension for faster algorithms.

Structural theorems and novel conditioning techniques improve analysis.

Abstract

The classical low rank approximation problem is to find a rank $k$ matrix $UV$ (where $U$ has $k$ columns and $V$ has $k$ rows) that minimizes the Frobenius norm of $A - UV$. Although this problem can be solved efficiently, we study an NP-hard variant of this problem that involves weights and regularization. A previous paper of [Razenshteyn et al. '16] derived a polynomial time algorithm for weighted low rank approximation with constant rank. We derive provably sharper guarantees for the regularized version by obtaining parameterized complexity bounds in terms of the statistical dimension rather than the rank, allowing for a rank-independent runtime that can be significantly faster. Our improvement comes from applying sharper matrix concentration bounds, using a novel conditioning technique, and proving structural theorems for regularized low rank problems.