Nearly Optimal Strong Coresets for $\ell_p$ Subspace Approximation

2026-08-26

AI Summary

该研究解决了$ ell_p$子空间逼近问题，通过改进采样和缩放方法降低了强核心集的大小，对于不同$p$值采用不同的技术手段。

Abstract

We study strong coresets for $\ell_p$ subspace approximation. Given $\A\in\R^{n\times d}$, the goal is to sample and rescale a small number of its rows to form $\Smat\A$ such that \[ \norm{\Smat\A(\I-\Pmat_F)}_{p,2}^p =(1\pm\eps)\norm{\A(\I-\Pmat_F)}_{p,2}^p \] simultaneously for every subspace $F\subseteq\R^d$ of dimension at most $k$, where $\Pmat_F$ is the orthogonal projector onto $F$. Woodruff and Yasuda (FOCS 2025)~\cite{WY25} obtained coreset sizes $\wtO_p(k\eps^{-4/p})$ for $1\leq p<2$ and $\wtO_p(k^{p/2}\eps^{-p})$ for $p>2$. We improve these bounds to $\wtO_p(k\eps^{-2})$ and $\wtO_p(k^{p/2}\eps^{-2})$, respectively. For $1\leq p<2$, our algorithm runs in $\wtO_p(\nnz(\A)+d^ω+k\eps^{-2})$ time. The resulting coreset size matches the known sampling lower bound~\cite{LWW21} up to logarithmic factors when $k+1\geq C\log(1/\eps)$ for an absolute constant $C$. For $p>2$, our algorithm runs in $\wtO_p(\nnz(\A)+d^ω)$ time, matching the running time of the Woodruff--Yasuda framework. We use different techniques in the two regimes. For $1\leq p<2$, we combine a bicriteria low-rank split with Lewis-weight sampling and empirical-process bounds independent of the output dimension. For $p>2$, we give a sharper analysis of the Woodruff--Yasuda construction. By retaining the truncation in its sampling probabilities throughout the row-count recurrence, we show that it achieves the improved $\eps^{-2}$ dependence.

Problem

• strong coresets
• ell_p subspace approximation
• subspace F
• projection error
• sampling and rescaling

Innovation

• strong coresets
• ell_p subspace approximation
• Lewis-weight sampling
• empirical-process bounds
• truncation in sampling probabilities