Slack and Margin Rescaling as Convex Extensions of Supermodular Functions

TL;DR

This paper introduces convex extensions of supermodular functions inspired by slack and margin rescaling techniques from structured SVMs, offering new analysis tools and algorithms for supermodular minimization with applications in image exemplar selection.

Contribution

It develops polynomial-time convex extensions of supermodular functions based on structured SVM surrogates, providing a new analysis framework and optimization approach.

Findings

Convex extensions can be tighter than existing surrogates.

The framework enables supermodular minimization via LP relaxations.

Application to image exemplar selection validates the theoretical methods.

Abstract

Slack and margin rescaling are variants of the structured output SVM, which is frequently applied to problems in computer vision such as image segmentation, object localization, and learning parts based object models. They define convex surrogates to task specific loss functions, which, when specialized to non-additive loss functions for multi-label problems, yield extensions to increasing set functions. We demonstrate in this paper that we may use these concepts to define polynomial time convex extensions of arbitrary supermodular functions, providing an analysis framework for the tightness of these surrogates. This analysis framework shows that, while neither margin nor slack rescaling dominate the other, known bounds on supermodular functions can be used to derive extensions that dominate both of these, indicating possible directions for defining novel structured output prediction surrogates. In addition to the analysis of structured prediction loss functions, these results imply an approach to supermodular minimization in which margin rescaling is combined with non-polynomial time convex extensions to compute a sequence of LP relaxations reminiscent of a cutting plane method. This approach is applied to the problem of selecting representative exemplars from a set of images, validating our theoretical contributions.