Sparsification of Decomposable Submodular Functions

TL;DR

This paper introduces a polynomial-time randomized sparsification algorithm for decomposable submodular functions, significantly reducing the number of functions needed for approximation while maintaining accuracy, with theoretical guarantees and empirical validation.

Contribution

The paper presents the first efficient sparsification method for decomposable submodular functions, with guarantees on approximation quality and independence from the original number of functions.

Findings

Algorithm achieves accurate approximation with fewer functions

Expected number of functions is independent of original count

Empirical results confirm theoretical effectiveness

Abstract

Submodular functions are at the core of many machine learning and data mining tasks. The underlying submodular functions for many of these tasks are decomposable, i.e., they are sum of several simple submodular functions. In many data intensive applications, however, the number of underlying submodular functions in the original function is so large that we need prohibitively large amount of time to process it and/or it does not even fit in the main memory. To overcome this issue, we introduce the notion of sparsification for decomposable submodular functions whose objective is to obtain an accurate approximation of the original function that is a (weighted) sum of only a few submodular functions. Our main result is a polynomial-time randomized sparsification algorithm such that the expected number of functions used in the output is independent of the number of underlying submodular functions in the original function. We also study the effectiveness of our algorithm under various constraints such as matroid and cardinality constraints. We complement our theoretical analysis with an empirical study of the performance of our algorithm.