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 ;-)

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One Response to New class of RIP matrices ?

  1. Igor Carron says:

    Laurent,

    Do you think that g could be behaving like the quantization function in the 1-bit compressive sensing approach of Petros ?

    http://www.merl.com/reports/docs/TR2010-014.pdf

    Cheers,

    Igor.

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