Online row sampling

16Citations
Citations of this article
17Readers
Mendeley users who have this article in their library.
Get full text

Abstract

Finding a small spectral approximation for a tall n × d matrix A is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of A. Row sampling improves interpretability, saves space when A is sparse, and preserves structure, which is important, e. g., when A represents a graph. However, correctly sampling rows from A can be costly when the matrix is large and cannot be stored and processed in memory. Hence, a number of recent publications focus on row sampling in the streaming setting, using little more space than what is required to store the returned approximation (Kelner–Levin, Theory Comput. Sys. 2013, Kapralov et al., SIAM J. Comp. 2017). Inspired by a growing body of work on online algorithms for machine learning and data analysis, we extend this work to a more restrictive online setting: we read rows of A one by one and immediately decide whether each row should be kept in the spectral approximation or discarded, without ever retracting these decisions. We present an extremely simple algorithm that approximates A up to multiplicative error 1 + ε and additive error δ using O(d log d log(ε‖A‖22/δ)/ε2 ) online samples, with memory overhead proportional to the cost of storing the spectral approximation. We also present an algorithm that uses O(d2 ) memory but only requires O(d log(ε‖A‖22/δ)/ε2 ) samples, which we show is optimal. Our methods are clean and intuitive, allow for lower memory usage than prior work, and expose new theoretical properties of leverage score based matrix approximation.

Cite

CITATION STYLE

APA

Cohen, M. B., Musco, C., & Pachocki, J. (2020). Online row sampling. Theory of Computing, 16. https://doi.org/10.4086/TOC.2020.V016A015

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free