We saw submodularity here before, take for instance Francis Bach's slides (page 99 and up) on Structured sparsity through convex optimization , where he mentions the use of submodularity to find new regularizers). Here is another way of using submodularity in compressive sensing: Map Estimation for Bayesian Mixture Models with Submodular Priors by Marwa El Halabi, Luca Baldassarre and Volkan Cevher

We propose a Bayesian approach where the signal structure can be represented by a mixture model with a submodular prior. We consider an observation model that leads to Lipschitz functions. Due to its combinatorial nature, computing the maximum a posteriori estimate for this model is NP-Hard, nonetheless our converging majorization-minimization scheme yields approximate estimates that, in practice, outperform state-of-the-art modular prior. We consider an observation model that leads to Lipschitz functions. Due to its combinatorial nature, computing the maximum a posteriori estimate for this model is NP-Hard, nonetheless our converging majorization-minimization scheme yields approximate estimates that, in practice, outperform state-of-the-art methods

why submodularity is a big deal, from the paper:

Submodularity is considered the discrete equivalent of convexity in the sense that submodular function minimization (SFM) admits efficient algorithms, with best known complexity of O(N5T+N6), where T is the function evaluation complexity [11]. In practice, however, the minimum-norm point algorithm is usually used whenever, the minimum-norm point algorithm is usually used, which commonly runs in O(N2), but has no known complexity [12]. Furthermore, for certain functions which are “graph representable” [13, 14], SFM is equivalent to the minimum s-t cut on an appropriate graph G(V,E), with time complexity 1 O(|E|min{|V|2/3,|E|1/2}) [15].

Relevant to the theme of submodularity are last year's Fourth Cargese Workshop on Combinatorial Optimization

## Bach:

## Iwata:

## McCormick:

## Naor:

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