CS: Bayesian Manifold based CS

Woohoo! As most of you know, one of the potentially interesting application of Compressive Sensing is the ability to perform detection and other operations in the Compressed Measurement world as opposed to having to reconstruct the signal. As shown by many, CS measurements can form a manifold that is close to the full fledged manifold. This was the subject of investigation of Mike Wakin in his thesis for instance. It just so happens that in the Bayesian Compressive Sensing page, I caught the following new text:

Manifold based CS theory has shown that if a signal lives in a low-dimensional manifold, then the signal can be reconstructed using only a few compressed measurements. However, till now there is no practical algorithm to implement CS on manifolds. Our recent work fills the gap by employing a nonparametric mixture of factor analyzers (MFA) to learn the manifold using training data, and then analytically reconstructing testing signals with compressed measurements. We also give bounds of the required number of measurements based on the concept of block-sparsity. The proposed methodology is validated on several synthetic and real datasets.

The paper is Compressive Sensing on Manifolds Using a Nonparametric Mixture of Factor Analyzers: Algorithm and Performance Bounds by Minhua Chen, Jorge Silva, John Paisley, Chunping Wang, David Dunson, and Lawrence Carin. The abstract reads:

Nonparametric Bayesian methods are employed to constitute a mixture of low-rank Gaussians, for data x element of R^N that are of high dimension N but are constrained to reside in a low-dimensional subregion of R^N. The number of mixture components and their rank are inferred automatically from the data. The resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds, based on compressive sensing (CS) projection measurements. The statistical CS inversion is performed analytically. We derive the required number of CS random measurements needed for successful reconstruction, based on easily computed quantities, drawing on block–sparsity properties. The proposed methodology is validated on several synthetic and real datasets.

And while I cannot seem to locate this MFA code within the Bayesian Compressive Sensing page, it is just a matter of time before it shows up. I note that the paper deals with smooth manifolds when sometimes these manifolds are non-differentiable.