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

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.

@article{li2025_2502.16912,
  title={ When Can We Solve the Weighted Low Rank Approximation Problem in Truly Subquadratic Time? },
  author={ Chenyang Li and Yingyu Liang and Zhenmei Shi and Zhao Song },
  journal={arXiv preprint arXiv:2502.16912},
  year={ 2025 }
}