Abstract: Spectral methods that leverage the singular value decomposition to reject noise and identify models are ubiquitous in signal processing. Despite their widespread adoption, little is understood about their finite-sample convergence properties. In this talk, I will provide a general framework, called atomic norm denoising, for understanding these spectral methods in the context of convex programming. The resulting optimization problems have generic, mean-squared-error guarantees and reduce to familiar soft-thresholding algorithms in the context of sparse approximation. I will specialize these techniques to provide a convex approach to spectrum estimation, estimating the frequencies and phases of a mixture of complex exponentials from noisy or missing data. These results serve as an extension of compressed sensing to the case when frequencies are not confined to lie on a discrete grid but can assume any value in a continuous interval. I will close with a discussion of other possible applications of this atomic-norm framework in statistical signal processing.
Joint work with Badri Bhaskar, Parikshit Shah, and Gongguo Tang
Bio: Benjamin Recht is an Assistant Professor in the Department of Computer Sciences at the University of Wisconsin-Madison and holds courtesy appointments in Electrical and Computer Engineering, Mathematics, and Statistics. He is a PI in the Wisconsin Institute for Discovery (WID), a newly founded center for research at the convergence of information technology, biotechnology, and nanotechnology. Ben received his B.S. in Mathematics from the University of Chicago, and received a M.S. and PhD from the MIT Media Laboratory. After completing his doctoral work, he was a postdoctoral fellow in the Center for the Mathematics of Information at Caltech. He is the recipient of an NSF Career Award, an Alfred P. Sloan Research Fellowship, and the 2012 SIAM/MOS Lagrange Prize in Continuous Optimization.