Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection

TL;DR

This paper analyzes greedy algorithms for subset selection and sparse approximation, introducing the submodularity ratio to explain their effectiveness even with highly correlated variables, and provides strong theoretical guarantees supported by experiments.

Contribution

It introduces the submodularity ratio as a key metric to analyze greedy algorithms' performance in correlated settings and improves approximation guarantees for subset and dictionary selection problems.

Findings

Greedy algorithms perform well due to the submodularity ratio.

The submodularity ratio predicts greedy performance better than spectral parameters.

Theoretical guarantees are strengthened for subset and dictionary selection.

Abstract

We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse approximation. We analyze the performance of widely used greedy heuristics, using insights from the maximization of submodular functions and spectral analysis. We introduce the submodularity ratio as a key quantity to help understand why greedy algorithms perform well even when the variables are highly correlated. Using our techniques, we obtain the strongest known approximation guarantees for this problem, both in terms of the submodularity ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further demonstrate the wide applicability of our techniques by analyzing greedy algorithms for the dictionary selection problem, and significantly improve the previously known guarantees. Our theoretical analysis is complemented by experiments on real-world and synthetic data sets; the experiments show that the submodularity ratio is a stronger predictor of the performance of greedy algorithms than other spectral parameters.