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Demixing refers to the challenge of identifying two structured signals given only the sum of the two signals and prior information about their structures. Examples include the problem of separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis, and the problem of decomposing an observed matrix into a low-rank matrix plus a sparse matrix. This paper describes and analyzes a framework, based on convex optimization, for solving these demixing problems, and many others. This work introduces a randomized signal model that ensures that the two structures are incoherent, i.e., generically oriented. For an observation from this model, this approach identifies a summary statistic that reflects the complexity of a particular signal. The difficulty of separating two structured, incoherent signals depends only on the total complexity of the two structures. Some applications include (1) demixing two signals that are sparse in mutually incoherent bases, (2) decoding spread-spectrum transmissions in the presence of impulsive errors, and (3) removing sparse corruptions from a low-rank matrix. In each case, the theoretical analysis of the convex demixing method closely matches its empirical behavior. [ABSTRACT FROM AUTHOR]

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can then be resolved using tools from Gaussian analysis. The approach leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality. [ABSTRACT FROM AUTHOR]

Abstract: An important problem in the theory of sparse approximation is to identify well-conditioned subsets of vectors from a general dictionary. In most cases, current results do not apply unless the number of vectors is smaller than the square root of the ambient dimension, so these bounds are too weak for many applications. This paper shatters the square-root bottleneck by focusing on random subdictionaries instead of arbitrary subdictionaries. It provides explicit bounds on the extreme singular values of random subdictionaries that hold with overwhelming probability. The results are phrased in terms of the coherence and spectral norm of the dictionary, which capture information about its global geometry. The proofs rely on standard tools from the area of Banach space probability. As an application, the paper shows that the conditioning of a subdictionary is the major obstacle to the uniqueness of sparse representations and the success of minimization techniques for signal recovery. Indeed, if a fixed subdictionary is well conditioned and its cardinality is slightly smaller than the ambient dimension, then a random signal formed from this subdictionary almost surely has no other representation that is equally sparse. Moreover, with overwhelming probability, the maximally sparse representation can be identified via minimization. Note that the results in this paper are not directly comparable with recent work on subdictionaries of random dictionaries. [Copyright &y& Elsevier]

This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which at- tempts to recover the ideal sparse signal by solving a convex pro- gram. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis. [ABSTRACT FROM AUTHOR]

Consider a data set of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called reaper, that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors and uses a relaxation of the set of orthogonal projectors to reach the convex formulation. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments that confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers lie near a low-dimensional subspace, there is a rigorous theory that describes when reaper can approximate this subspace. [ABSTRACT FROM AUTHOR]