Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components

TL;DR

This paper introduces fast approximation algorithms for minimizing sums of cardinality-based submodular functions, improving efficiency and applicability in image segmentation and hypergraph clustering tasks.

Contribution

It presents the first approximation algorithms for cardinality-based submodular minimization using sparse graph reductions and piecewise linear approximations.

Findings

Algorithms achieve faster runtimes than previous methods.

Practical improvements demonstrated on image segmentation and hypergraph clustering.

Theoretical analysis confirms efficiency gains.

Abstract

Minimizing a sum of simple submodular functions of limited support is a special case of general submodular function minimization that has seen numerous applications in machine learning. We develop fast techniques for instances where components in the sum are cardinality-based, meaning they depend only on the size of the input set. This variant is one of the most widely applied in practice, encompassing, e.g., common energy functions arising in image segmentation and recent generalized hypergraph cut functions. We develop the first approximation algorithms for this problem, where the approximations can be quickly computed via reduction to a sparse graph cut problem, with graph sparsity controlled by the desired approximation factor. Our method relies on a new connection between sparse graph reduction techniques and piecewise linear approximations to concave functions. Our sparse reduction technique leads to significant improvements in theoretical runtimes, as well as substantial practical gains in problems ranging from benchmark image segmentation tasks to hypergraph clustering problems.