Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components
This work addresses a widely applied variant of submodular function minimization for tasks such as image segmentation and hypergraph clustering, offering incremental improvements through new approximation techniques.
The paper tackled the problem of minimizing sums of cardinality-based submodular functions, which is common in applications like image segmentation and hypergraph clustering, by developing the first approximation algorithms that reduce it to sparse graph cuts, leading to significant theoretical runtime improvements and practical gains.
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.