Reflection methods for user-friendly submodular optimization

TL;DR

This paper introduces a new reflection-based optimization method for submodular function minimization that is efficient, easy to implement, and applicable to various machine learning tasks.

Contribution

It presents a novel, non-approximate, and parameter-free approach that leverages decomposability and continuous approximation for submodular minimization.

Findings

Method is easy to implement and parallelize.

Effective on image segmentation tasks.

Solves both continuous and discrete problems.

Abstract

Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision. Consequently, there is need for efficient optimization procedures for submodular functions, especially for minimization problems. While general submodular minimization is challenging, we propose a new method that exploits existing decomposability of submodular functions. In contrast to previous approaches, our method is neither approximate, nor impractical, nor does it need any cumbersome parameter tuning. Moreover, it is easy to implement and parallelize. A key component of our method is a formulation of the discrete submodular minimization problem as a continuous best approximation problem that is solved through a sequence of reflections, and its solution can be easily thresholded to obtain an optimal discrete solution. This method solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruction. In our experiments, we illustrate the benefits of our method on two image segmentation tasks.