Abstract: Compressed Sensing, originally an idea based on theory of high-dimensional convex geometry, eventually became an industry standard for rapid MR Imaging with implementations by all three major MRI device manufacturers. The original ideas were based on scalar measurements. Recent trends, for example in receiver coil arrays, point towards vector observables. 

We review some Compressed Sensing theory through the lens of Approximate Message Passing (AMP) which shows how an elementary denoiser can be turned into a procedure for a drastically more ambitious problem of compressed sensing with complete theoretical understanding of reconstruction accuracy and other properties.  In the case of vector observables, the simple procedure is James-Stein shrinkage; with the help of AMP, it becomes a procedure of multiple-measurement vector compressed sensing.

State Evolution analysis of the sparsity-undersampling phase diagram shows that with high-dimensional measurement vectors, James-Stein Compressed sensing has a theoretically unimprovable phase diagram, and empirically works near-optimally even in low vector dimensions. In particular, this is far better than known convex optimization approaches.

This is joint work with Apratim Dey (Stanford Statistics)

Bio

David Donoho is the Anne T. and Robert M. Bass Professor of Humanities and Sciences Professor of Statistics. He coined the notion of compressed sensing, which has impacted many scientific and technical fields, including magnetic resonance imaging in medicine, where it has been implemented in FDA-approved medical imaging protocols and is already used in millions of actual patient MRIs.

In recent years Donoho and his postdocs and students have been studying large-scale covariance matrix estimation, large-scale matrix denoising, detection of rare and weak signals among many pure noise non-signals, compressed sensing and related scientific imaging problems, and most recently, empirical deep learning.