To Convexify or Not? Regression with Clustering Penalties on Graphs

We consider minimization problems that are compositions of convex functions of a vector $\x \in\reals^N$ with submodular set functions of its support (i.e., indices of the non-zero coefficients of $\x$). Such problems are in general difficult for large $N$ due to their combinatorial nature. In this setting, existing approaches rely on ``convexifications" of the submodular set function based on the Lov'asz extension for tractable approximations. In this paper, we first demonstrate that such convexifications can fundamentally change the nature of the underlying submodular regularization. We then provide a majorization-minimization framework for the minimization of such composite objectives. For concreteness, we use the Ising model to motivate a submodular regularizer, establish the total variation semi-norm as its Lov'asz extension, and numerically illustrate our new optimization framework.