The Summer School "Structured Regularization for High-Dimensional Data Analysis" is on-going in Paris right now at IHP Paris. Here are some of the videos of the second and third day (Tuesday and Wednesday) with Anders Hansen , Andrea Montanari, Francis Bach

__and__Carlos Fernandez-GrandaOn foundational computation problems in l1 and TV regularization

A.Hansen - 3/4 - 20/06/2017

Computing the Non-computable via sparsity

On foundational computation barriers in l1 and TV regularization

A.Hansen - 4/4 - 20/06/2017

Abstract: Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the submodular set-function to a convex function, which opens up tools from convex optimization. Submodularity goes beyond set-functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains, where it corresponds essentially to cross second-derivatives being nonpositive. In this talk, I will show that most results relating submodularity and convexity for set-functions can be extended to all submodular functions. In particular, (a) I will naturally define a continuous extension in a set of probability measures, (b) show that the extension is convex if and only if the original function is submodular, (c) prove that the problem of minimizing a submodular function is equivalent to a typically non-smooth convex optimization problem. Most of these extensions from the set-function situation are obtained by drawing links with the theory of multi-marginal optimal transport, which provides also a new interpretation of existing results for set-functions. I will then provide practical algorithms to minimize generic submodular functions on discrete domains, with associated convergence rates, and an application to proximal operators for non-convex penalty functions. Preprint available here.

Abstract: In the 70s and 80s geophysicists proposed using l1-norm regularization for deconvolution problem in the context of reflection seismology. Since then such methods have had a great impact in high-dimensional statistics and in signal-processing applications, but until recently their performance on the original deconvolution problem was not well understood theoretically. In this talk we provide an analysis of optimization-based methods for the deconvolution problem, including results on irregular sampling and sparse corruptions that highlight the modeling flexibility of these techniques.

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