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We consider a class of sparsity-inducing regularization terms based onsubmodular functions. While previous work has focused on non-decreasingfunctions, we explore symmetric submodular functions and their \lovaextensions. We show that the Lovasz extension may be seen as the convexenvelope of a function that depends on level sets (i.e., the set of indiceswhose corresponding components of the underlying predictor are greater than agiven constant): this leads to a class of convex structured regularizationterms that impose prior knowledge on the level sets, and not only on thesupports of the underlying predictors. We provide a unified set of optimizationalgorithms, such as proximal operators, and theoretical guarantees (allowedlevel sets and recovery conditions). By selecting specific submodularfunctions, we give a new interpretation to known norms, such as the totalvariation; we also define new norms, in particular ones that are based on orderstatistics with application to clustering and outlier detection, and on noisycuts in graphs with application to change point detection in the presence ofoutliers.

Shaping Level Sets with Submodular Functions