We often model signals from the physical world with continuous parameterizations. Unfortunately, continuous models pose problems for the tools of sparse approximation, and popular discrete approximations are fraught with theoretical and algorithmic issues. In this talk, I will propose a general, convex-optimization framework—called atomic-norm denoising—that obviates discretization and gridding by generalizing sparse approximation to continuous dictionaries.

As an extended example that highlights the salient features of the atomic-norm framework, I will highlight the problem of estimating the frequencies and phases of a mixture of complex exponentials from noisy, incomplete data. I will demonstrate that atomic-norm denoising outperforms state-of-the-art spectral and sparse-approximation methods in both theory and practice. I will then close with a discussion of additional applications of the atomic-norm framework including deconvolution, deblurring, and system identification.

Joint work with Badri Bhaskar, Parikshit Shah, and Gongguo Tang.