Conditions for Existence of Dual Certificates in Rank-One Semidefinite Problems

Several signal recovery tasks can be relaxed into semidefinite programs with rank-one minimizers. A common technique for proving these programs succeed is to construct a dual certificate. Unfortunately, dual certificates may not exist under some formulations of semidefinite programs. In order to put problems into a form where dual certificate arguments are possible, it is important to develop conditions under which the certificates exist. In this paper, we provide an example where dual certificates do not exist. We then present a completeness condition under which they are guaranteed to exist. For programs that do not satisfy the completeness condition, we present a completion process which produces an equivalent program that does satisfy the condition. The important message of this paper is that dual certificates may not exist for semidefinite programs that involve orthogonal measurements with respect to positive-semidefinite matrices. Such measurements can interact with the positive-semidefinite constraint in a way that implies additional linear measurements. If these additional measurements are not included in the problem formulation, then dual certificates may fail to exist. As an illustration, we present a semidefinite relaxation for the task of finding the sparsest element in a subspace. One formulation of this program does not admit dual certificates. The completion process produces an equivalent formulation which does admit dual certificates

The achievable performance of convex demixing

Demixing is the problem of identifying multiple structured signals from a superimposed, undersampled, and noisy observation. This work analyzes a general framework, based on convex optimization, for solving demixing problems. When the constituent signals follow a generic incoherence model, this analysis leads to precise recovery guarantees. These results admit an attractive interpretation: each signal possesses an intrinsic degrees-of-freedom parameter, and demixing can succeed if and only if the dimension of the observation exceeds the total degrees of freedom present in the observation

Efficient independent component analysis

Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on M-estimates have been proposed for estimating the mixing matrix. Recently, several nonparametric methods have been developed, but in-depth analysis of asymptotic efficiency has not been available. We analyze ICA using semiparametric theories and propose a straightforward estimate based on the efficient score function by using B-spline approximations. The estimate is asymptotically efficient under moderate conditions and exhibits better performance than standard ICA methods in a variety of simulations.Comment: Published at http://dx.doi.org/10.1214/009053606000000939 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org