Spectral Guarantees for Adversarial Streaming PCA
Eric Price, Zhiyang Xun

TL;DR
This paper establishes spectral ratio thresholds for effective adversarial streaming PCA, introduces a new analysis of Oja's algorithm, and demonstrates fundamental space complexity limits for the problem.
Contribution
It provides the first spectral-tailored analysis of Oja's algorithm for adversarial streams and identifies necessary spectral ratio bounds for streaming PCA.
Findings
Mergeable summaries require spectral ratio R = Ω(√d).
A variant of Oja's algorithm achieves low error with R = O(log n log d).
No algorithms with sub-quadratic space can succeed at R = O(1).
Abstract
In streaming PCA, we see a stream of vectors and want to estimate the top eigenvector of their covariance matrix. This is easier if the spectral ratio is large. We ask: how large does need to be to solve streaming PCA in space? Existing algorithms require . We show: (1) For all mergeable summaries, is necessary. (2) In the insertion-only model, a variant of Oja's algorithm gets error for . (3) No algorithm with space gets error for . Our analysis is the first application of Oja's algorithm to adversarial streams. It is also the first algorithm for adversarial streaming PCA that is designed for a spectral, rather than Frobenius, bound on the tail; and the bound it needs is…
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Taxonomy
TopicsSparse and Compressive Sensing Techniques · Security in Wireless Sensor Networks · Anomaly Detection Techniques and Applications
MethodsPrincipal Components Analysis
