When Can We Solve the Weighted Low Rank Approximation Problem in Truly Subquadratic Time?

• URL: http://arxiv.org/abs/2502.16912v1
• Date: Mon, 24 Feb 2025 07:18:24 GMT
• Title: When Can We Solve the Weighted Low Rank Approximation Problem in Truly Subquadratic Time?
• Authors: Chenyang Li, Yingyu Liang, Zhenmei Shi, Zhao Song,
• Abstract summary: The goal is to find two low-rank matrices $U, V in mathbbRn times k$ such that the cost of $| W circ (U Vtop - A) |_F2$ is minimized.<n>In this work, we show that there is a certain regime, even if $A$ and $W$ are dense, we can still hope to solve the weighted low-rank approximation problem in almost linear $n1+o(1)$ time.
• Score: 22.047262762274414
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The weighted low-rank approximation problem is a fundamental numerical linear algebra problem and has many applications in machine learning. Given a $n \times n$ weight matrix $W$ and a $n \times n$ matrix $A$, the goal is to find two low-rank matrices $U, V \in \mathbb{R}^{n \times k}$ such that the cost of $\| W \circ (U V^\top - A) \|_F^2$ is minimized. Previous work has to pay $\Omega(n^2)$ time when matrices $A$ and $W$ are dense, e.g., having $\Omega(n^2)$ non-zero entries. In this work, we show that there is a certain regime, even if $A$ and $W$ are dense, we can still hope to solve the weighted low-rank approximation problem in almost linear $n^{1+o(1)}$ time.

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