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Shaping Level Sets with Submodular Functions
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@MISC{Bach_shapinglevel,
author = {Francis Bach},
title = {Shaping Level Sets with Submodular Functions},
year = {}
}
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Abstract
We consider a class of sparsity-inducing regularization terms based on submodular functions. While previous work has focused on non-decreasing functions, we explore symmetric submodular functions and their Lovász extensions. We show that the Lovász extension may be seen as the convex envelope of a function that depends on level sets (i.e., the set of indices whose corresponding components of the underlying predictor are greater than a given constant): this leads to a class of convex structured regularization terms that impose prior knowledge on the level sets, and not only on the supports of the underlying predictors. We provide unified optimization algorithms, such as proximal operators, and theoretical guarantees (allowed level sets and recovery conditions). By selecting specific submodular functions, we give a new interpretation to known norms, such as the total variation; we also define new norms, in particular ones that are based on order statistics with application to clustering and outlier detection, and on noisy cuts in graphs with application to change point detection in the presence of outliers. 1
Keyphrases
level set submodular function lov sz extension underlying predictor non-decreasing function convex envelope sparsity-inducing regularization term new interpretation order statistic proximal operator regularization term previous work new norm total variation specific submodular function prior knowledge noisy cut unified optimization algorithm theoretical guarantee recovery condition point detection symmetric submodular function particular one
