Convexity in source separation: Models, geometry, and algorithms

Source separation or demixing is the process of extracting multiple components entangled within a signal. Contemporary signal processing presents a host of difficult source separation problems, from interference cancellation to background subtraction, blind deconvolution, and even dictionary learning. Despite the recent progress in each of these applications, advances in high-throughput sensor technology place demixing algorithms under pressure to accommodate extremely high-dimensional signals, separate an ever larger number of sources, and cope with more sophisticated signal and mixing models. These difficulties are exacerbated by the need for real-time action in automated decision-making systems. Recent advances in convex optimization provide a simple framework for efficiently solving numerous difficult demixing problems. This article provides an overview of the emerging field, explains the theory that governs the underlying procedures, and surveys algorithms that solve them efficiently. We aim to equip practitioners with a toolkit for constructing their own demixing algorithms that work, as well as concrete intuition for why they work

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric $K$-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known {\em a priori} to lie in certain frequency bands. Numerical examples are also provided.Comment: 23 pages, accepted by Signal Processin

Joint Sparsity Recovery for Spectral Compressed Sensing

Compressed Sensing (CS) is an effective approach to reduce the required number of samples for reconstructing a sparse signal in an a priori basis, but may suffer severely from the issue of basis mismatch. In this paper we study the problem of simultaneously recovering multiple spectrally-sparse signals that are supported on the same frequencies lying arbitrarily on the unit circle. We propose an atomic norm minimization problem, which can be regarded as a continuous counterpart of the discrete CS formulation and be solved efficiently via semidefinite programming. Through numerical experiments, we show that the number of samples per signal may be further reduced by harnessing the joint sparsity pattern of multiple signals