Sketching Matrix Least Squares via Leverage Scores Estimates
(2201.10638)Abstract
We consider the matrix least squares problem of the form $| \mathbf{A} \mathbf{X}-\mathbf{B} |F2$ where the design matrix $\mathbf{A} \in \mathbb{R}{N \times r}$ is tall and skinny with $N \gg r$. We propose to create a sketched version $| \tilde{\mathbf{A}}\mathbf{X}-\tilde{\mathbf{B}} |F2$ where the sketched matrices $\tilde{\mathbf{A}}$ and $\tilde{\mathbf{B}}$ contain weighted subsets of the rows of $\mathbf{A}$ and $\mathbf{B}$, respectively. The subset of rows is determined via random sampling based on leverage score estimates for each row. We say that the sketched problem is $\epsilon$-accurate if its solution $\tilde{\mathbf{X}}{\rm \text{opt}} = \text{argmin } | \tilde{\mathbf{A}}\mathbf{X}-\tilde{\mathbf{B}} |F2$ satisfies $|\mathbf{A}\tilde{\mathbf{X}}{\rm \text{opt}}-\mathbf{B} |F2 \leq (1+\epsilon) \min | \mathbf{A}\mathbf{X}-\mathbf{B} |_F2$ with high probability. We prove that the number of samples required for an $\epsilon$-accurate solution is $O(r/(\beta \epsilon))$ where $\beta \in (0,1]$ is a measure of the quality of the leverage score estimates.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.