Learning smooth models of nonsmooth functions via convex optimization

International audienceThis paper proposes a learning framework and a set of algorithms for nonsmooth regression, i.e., for learning piecewise smooth target functions with discontinuities in the function itself or the derivatives at unknown locations. In the proposed approach, the model belongs to a class of smooth functions. Though constrained to be globally smooth, the trained model can have very large derivatives at particular locations to approximate the nonsmoothness of the target function. This is obtained through the definition of new regularization terms which penalize the derivatives in a location-dependent manner and training algorithms in the form of convex optimization problems. Examples of application to hybrid dynamical system identification and image reconstruction are provided

Reflection methods for user-friendly submodular optimization

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.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013