8,838 research outputs found

    Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization

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    In this paper, we propose an inexact block coordinate descent algorithm for large-scale nonsmooth nonconvex optimization problems. At each iteration, a particular block variable is selected and updated by inexactly solving the original optimization problem with respect to that block variable. More precisely, a local approximation of the original optimization problem is solved. The proposed algorithm has several attractive features, namely, i) high flexibility, as the approximation function only needs to be strictly convex and it does not have to be a global upper bound of the original function; ii) fast convergence, as the approximation function can be designed to exploit the problem structure at hand and the stepsize is calculated by the line search; iii) low complexity, as the approximation subproblems are much easier to solve and the line search scheme is carried out over a properly constructed differentiable function; iv) guaranteed convergence of a subsequence to a stationary point, even when the objective function does not have a Lipschitz continuous gradient. Interestingly, when the approximation subproblem is solved by a descent algorithm, convergence of a subsequence to a stationary point is still guaranteed even if the approximation subproblem is solved inexactly by terminating the descent algorithm after a finite number of iterations. These features make the proposed algorithm suitable for large-scale problems where the dimension exceeds the memory and/or the processing capability of the existing hardware. These features are also illustrated by several applications in signal processing and machine learning, for instance, network anomaly detection and phase retrieval

    DOLPHIn - Dictionary Learning for Phase Retrieval

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    We propose a new algorithm to learn a dictionary for reconstructing and sparsely encoding signals from measurements without phase. Specifically, we consider the task of estimating a two-dimensional image from squared-magnitude measurements of a complex-valued linear transformation of the original image. Several recent phase retrieval algorithms exploit underlying sparsity of the unknown signal in order to improve recovery performance. In this work, we consider such a sparse signal prior in the context of phase retrieval, when the sparsifying dictionary is not known in advance. Our algorithm jointly reconstructs the unknown signal - possibly corrupted by noise - and learns a dictionary such that each patch of the estimated image can be sparsely represented. Numerical experiments demonstrate that our approach can obtain significantly better reconstructions for phase retrieval problems with noise than methods that cannot exploit such "hidden" sparsity. Moreover, on the theoretical side, we provide a convergence result for our method

    PhasePack: A Phase Retrieval Library

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    Phase retrieval deals with the estimation of complex-valued signals solely from the magnitudes of linear measurements. While there has been a recent explosion in the development of phase retrieval algorithms, the lack of a common interface has made it difficult to compare new methods against the state-of-the-art. The purpose of PhasePack is to create a common software interface for a wide range of phase retrieval algorithms and to provide a common testbed using both synthetic data and empirical imaging datasets. PhasePack is able to benchmark a large number of recent phase retrieval methods against one another to generate comparisons using a range of different performance metrics. The software package handles single method testing as well as multiple method comparisons. The algorithm implementations in PhasePack differ slightly from their original descriptions in the literature in order to achieve faster speed and improved robustness. In particular, PhasePack uses adaptive stepsizes, line-search methods, and fast eigensolvers to speed up and automate convergence

    Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

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    Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine - change of title in the last revisio
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