5,099 research outputs found

    Simultaneously Structured Models with Application to Sparse and Low-rank Matrices

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    The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and low-rank. Often norms that promote each individual structure are known, and allow for recovery using an order-wise optimal number of measurements (e.g., â„“1\ell_1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multi-objective optimization with these norms, then we can do no better, order-wise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and low-rank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the â„“1\ell_1 and nuclear norms requires many more measurements. This proves an order-wise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and low-rank tensor completion.Comment: 38 pages, 9 figure

    Undersampled Phase Retrieval with Outliers

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    We propose a general framework for reconstructing transform-sparse images from undersampled (squared)-magnitude data corrupted with outliers. This framework is implemented using a multi-layered approach, combining multiple initializations (to address the nonconvexity of the phase retrieval problem), repeated minimization of a convex majorizer (surrogate for a nonconvex objective function), and iterative optimization using the alternating directions method of multipliers. Exploiting the generality of this framework, we investigate using a Laplace measurement noise model better adapted to outliers present in the data than the conventional Gaussian noise model. Using simulations, we explore the sensitivity of the method to both the regularization and penalty parameters. We include 1D Monte Carlo and 2D image reconstruction comparisons with alternative phase retrieval algorithms. The results suggest the proposed method, with the Laplace noise model, both increases the likelihood of correct support recovery and reduces the mean squared error from measurements containing outliers. We also describe exciting extensions made possible by the generality of the proposed framework, including regularization using analysis-form sparsity priors that are incompatible with many existing approaches.Comment: 11 pages, 9 figure

    Balancing Sparsity and Rank Constraints in Quadratic Basis Pursuit

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    We investigate the methods that simultaneously enforce sparsity and low-rank structure in a matrix as often employed for sparse phase retrieval problems or phase calibration problems in compressive sensing. We propose a new approach for analyzing the trade off between the sparsity and low rank constraints in these approaches which not only helps to provide guidelines to adjust the weights between the aforementioned constraints, but also enables new simulation strategies for evaluating performance. We then provide simulation results for phase retrieval and phase calibration cases both to demonstrate the consistency of the proposed method with other approaches and to evaluate the change of performance with different weights for the sparsity and low rank structure constraints
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