An Online Sparsification Algorithm from the Book

2026-07-25

AI Summary

This work investigates the robustness of online spectral sparsification algorithms under adaptive data streams, where incoming rows may depend on the algorithm’s prior random choices. We establish, for the first time, that the original online leverage score sampling algorithm is resilient against adaptive adversaries. To achieve this, we introduce a Freedman-type matrix martingale inequality combined with a dynamically evolving isotropic embedding. This yields the first online sparsification algorithm provably effective for adaptive streams. In the context of graph sparsification, our method runs in nearly linear time with respect to the stream size, produces a sparsifier of size nearly matching the information-theoretic optimum $O(d\varepsilon^{-2}\log^2 d)$, and incurs memory usage proportional to the size of the output sparsifier.

Abstract

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.

Problem

• online sparsification
• adaptive streams
• spectral approximation
• leverage scores
• matrix martingale

Innovation

• online spectral sparsification
• adaptive streams
• leverage score sampling
• matrix martingale inequality
• dynamic isotropic map