I have studied the exploitation of sparse signals in signal recovery, including for denoising, superresolution, and solution of underdetermined equations. This research with collaborators showed that ℓ¹ penalization was an effective and even optimal way to exploit sparsity of the object to be recovered.

Compressed sensing has impacted scientific and technical fields, including magnetic resonance imaging in medicine, where it has been implemented in FDA-approved medical imaging protocols already used for millions of patient MRIs.

In recent years, my 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.