A large number of problems in optimization, machine learning, signal processing can be effectively addressed by suitable semidefinite programming (SDP) relaxations. Unfortunately, generic SDP solvers hardly scale beyond instances with a few hundreds variables (in the underlying combinatorial problem). On the other hand, it has been observed empirically that an effective strategy amounts to introducing a (non-convex) rank constraint, and solving the resulting smooth optimization problem by ascent methods. This non-convex problem has --generically-- a large number of local maxima, and the reason for this success is therefore unclear. This paper provides rigorous support for this approach. For the problem of maximizing a linear functional over the elliptope, we prove that all local maxima are within a small gap from the SDP optimum. In several problems of interest, arbitrarily small relative error can be achieved by taking the rank constraint @math to be of order one, independently of the problem size.
Bandeira, Boumal and Voroninski recently considered the extreme case @math , in the specific example of the @math synchronization problem @cite_11 . They proved that, if the signal is strong enough, then this approach can effectively recover the underlying signal. Namely, all local minima are correlated with the signal.
@cite_11
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Suppose we wish to recover a signal amssym @math from m intensity measurements of the form , ; that is, from data in which phase information is missing. We prove that if the vectors are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program–-a trace-norm minimization problem; this holds with large probability provided that m is on the order of , and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis-a-vis additive noise. © 2012 Wiley Periodicals, Inc. Recovering an unknown complex signal from the magnitude of linear combinations of the signal is referred to as phase retrieval. We present an exact performance analysis of a recently proposed convex-optimization-formulation for this problem, known as PhaseMax. Standard convex-relaxation-based methods in phase retrieval resort to the idea of “lifting” which makes them computationally inefficient, since the number of unknowns is effectively squared. In contrast, PhaseMax is a novel convex relaxation that does not increase the number of unknowns. Instead it relies on an initial estimate of the true signal which must be externally provided. In this paper, we investigate the required number of measurements for exact recovery of the signal in the large system limit and when the linear measurement matrix is random with iid standard normal entries. If @math denotes the dimension of the unknown complex signal and @math the number of phaseless measurements, then in the large system limit, @math measurements is necessary and sufficient to recover the signal with high probability, where @math is the angle between the initial estimate and the true signal. Our result indicates a sharp phase transition in the asymptotic regime which matches the empirical result in numerical simulations. This paper studies the problem of recovering a sparse signal x ∈ ℝn from highly corrupted linear measurements y = Ax + e ∈ ℝm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any sufficiently sparse signal x can be recovered by solving an l1 -minimization problem min ||x||1 + ||e||1 subject to y = Ax + e. More precisely, if the fraction of the support of the error e is bounded away from one and the support of a: is a very small fraction of the dimension m, then as m becomes large the above l1 -minimization succeeds for all signals x and almost all sign-and-support patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100 of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of independent identically distributed (i.i.d.) Gaussian vectors with nonzero mean and small variance, dubbed the "cross-and-bouquet" (CAB) model. Simulations and experiments corroborate the findings, and suggest extensions to the result. The problem of signal recovery from the autocorrelation, or equivalently, the magnitudes of the Fourier transform, is of paramount importance in various fields of engineering. In this work, for one-dimensional signals, we give conditions, which when satisfied, allow unique recovery from the autocorrelation with very high probability. In particular, for sparse signals, we develop two non-iterative recovery algorithms. One of them is based on combinatorial analysis, which we prove can recover signals upto sparsity o(n1 3) with very high probability, and the other is developed using a convex optimization based framework, which numerical simulations suggest can recover signals upto sparsity o(n1 2) with very high probability.