Online Spectral Approximation in Random Order Streams

TL;DR

This paper introduces improved online algorithms for spectral approximation of positive semidefinite matrices, achieving better bounds without additive errors and analyzing the random order setting with new lower bounds.

Contribution

It presents an online spectral approximation algorithm that avoids additive errors and offers tighter bounds for small-rank matrices, along with algorithms for the random order setting and new lower bounds.

Findings

New online algorithm avoids additive error with similar complexity.

Algorithms efficiently handle the random order setting.

Lower bounds on approximation size are established for the online random order setting.

Abstract

This paper studies spectral approximation for a positive semidefinite matrix in the online setting. It is known in [Cohen et al. APPROX 2016] that we can construct a spectral approximation of a given $n \times d$ matrix in the online setting if an additive error is allowed. In this paper, we propose an online algorithm that avoids an additive error with the same time and space complexities as the algorithm of Cohen et al., and provides a better upper bound on the approximation size when a given matrix has small rank. In addition, we consider the online random order setting where a row of a given matrix arrives uniformly at random. In this setting, we propose time and space efficient algorithms to find a spectral approximation. Moreover, we reveal that a lower bound on the approximation size in the online random order setting is $\Omega (d \epsilon^{-2} \log n)$, which is larger than the one in the offline setting by an $\mathrm{O}\left( \log n \right)$ factor.