# New class of RIP matrices ?

Wouaw, almost one year and half without any post here…. Shame on me! I’ll try to be more productive with shorter posts now ;-) I just found this interesting paper about concentration properties of submodular function (very common in “Graph Cut” methods for instance) on arxiv:

### A note on concentration of submodular functions. (arXiv:1005.2791v1 [cs.DM])

Jan Vondrak, May 18, 2010 We survey a few concentration inequalities for submodular and fractionally subadditive functions of independent random variables, implied by the entropy method for self-bounding functions. The power of these concentration bounds is that they are dimension-free, in particular implying standard deviation O(\sqrt{\E[f]}) rather than O(\sqrt{n}) which can be obtained for any 1-Lipschitz function of n variables.

In particular, the author shows some interesting concentration results in his corollary 3.2.

Without having performed any developments, I’m wondering if this result could serve to define a new class of matrices (or non-linear operators) satisfying either the Johnson-Lindenstrauss Lemma or the Restricted Isometry Property. For instance, by starting from Bernoulli vectors $X$, i.e., the rows of a sensing matrix, and defining some specific submodular (or self-bounding) functions $f$ (e.g. $f(X_1,\cdots, X_n) = g(X^T a)$ for some sparse vector $a$ and a “kind” function $g$), I’m wondering if the concentration results above are better than those coming from the classical concentration inequalities (based on the Lipschitz properties of $f$ or $g$. See e.g., the books of Ledoux and Talagrand)? Ok, all this is perhaps just due to too early thoughts …. before my mug of black coffee ;-)