书中的一种在线稀疏化算法
An Online Sparsification Algorithm from the Book
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中文总结 AI 辅助
研究自适应流中在线谱稀疏化算法是否有效，提出带演变各向同性映射的Freedman型矩阵鞅不等式，证明原始算法对自适应对手稳健，给出近最优大小且工作内存与稀疏化矩阵大小成比例的算法，谱图稀疏化时有接近线性时间实现。
中文摘要 AI 辅助
在其开创性论文[Cohen等人，2016]中，Cohen、Musco和Pachocki提出了一种自然且简单的在线谱稀疏化算法：矩阵\(A\)的行\(a_1,a_2,\ldots\in\mathbb{R}^d\)逐个到达，当行\(a_i\)到达时，以与其当前杠杆得分\(\tau^{\mathrm{OL}}(a_i)=a_i^\top(A_i^\top A_i)^\dagger a_i\)（其中\(A_i = [a_1, a_2, \ldots, a_i]^\top\)）成比例的概率（在适当重新加权后）附加到稀疏化矩阵\(\tilde{A}\)上，否则永远丢弃。对于遗忘流，他们表明这能以\(O(d\epsilon^{-2}\log^2 d)\)行保持每个\(A\)的\((1\pm\epsilon)\)谱近似\(\tilde{A}\)。一个自然的问题是该算法对自适应流是否有效，原始证明不能直接扩展。本文表明原始在线杠杆得分采样算法对自适应对手确实稳健。主要技术贡献是一个带有不断演变的各向同性映射的Freedman型矩阵鞅不等式，从而给出了第一个用于自适应流的在线稀疏化算法，其稀疏化矩阵大小接近最优\(O(d \varepsilon^{-2}\log^2 d)\)，工作内存与稀疏化矩阵大小成比例。对于谱图稀疏化的特殊情况，还提供了一种在流大小上接近线性时间运行的实现。
英文摘要
In their seminal paper [Cohen et al., 2016], Cohen, Musco, and Pachocki proposed a natural and simple online spectral sparsification algorithm: rows $a_1, a_2, \ldots \in \mathbb{R}^d$ of a matrix $A$ arrive one-by-one, and when row $a_i$ arrives, it is appended to sparsifier $\tilde{A}$ (after appropriately reweighting it) with probability proportional to its current leverage score $$ τ^{\mathrm{OL}}(a_i)=a_i^\top(A_i^\top A_i)^\dagger a_i, \text{ where }A_i = [a_1, a_2, \ldots, a_i]^\top $$ or otherwise discarded forever. For oblivious streams, they showed that this maintains a $(1\pmε)$-spectral approximation $\tilde{A}$ of every $A$ with $O(dε^{-2}\log^2 d)$ many rows. A natural question is whether the same algorithm works for adaptive streams, where each row may depend on the algorithm's previous random choices. The original proof does not extend directly: it analyzes the process in isotropic position with respect to the final matrix $A$, which is not fixed in advance under adaptivity. As an extension of this proof framework remained elusive, various algorithmic variants have since been suggested. In this paper, we show that the original online leverage-score sampling algorithm is indeed robust to adaptive adversaries. Our main technical contribution is a Freedman-type matrix martingale inequality with an evolving isotropic map, allowing the isotropic map used in the concentration argument to change with the stream. As a consequence, this gives the first online sparsification algorithm for adaptive streams that yields a sparsifier of near-optimal size $O(d \varepsilon^{-2}\log^2 d)$ whose working memory is proportional to the size of the sparsifier. For the special case of spectral graph sparsification, we provide an implementation that additionally runs in time near-linear in the stream size.
