Online Row Sampling

TL;DR

This paper introduces simple online algorithms for row sampling to produce spectral approximations of large matrices with minimal memory, suitable for streaming data, and proves their optimality.

Contribution

It presents novel online algorithms for row sampling that operate with low memory and provide provably optimal spectral approximations of matrices.

Findings

The algorithms achieve spectral approximation with low sample complexity.

Memory usage can be reduced to near the size of the spectral approximation.

The methods reveal new theoretical insights into leverage score based matrix approximation.

Abstract

Finding a small spectral approximation for a tall $n \times d$ 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 row structure, which is especially important, for example, 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 outputted approximation [KL13, KLM+14]. 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 $\epsilon$ and additive error $\delta$ using $O(d \log d \log(\epsilon||A||_2/\delta)/\epsilon^2)$ online samples, with memory overhead proportional to the cost of storing the spectral approximation. We also present an algorithm that uses $O(d^2$) memory but only requires $O(d\log(\epsilon||A||_2/\delta)/\epsilon^2)$ 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.