3,494 research outputs found

    Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

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    A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

    Sample Complexity Analysis for Learning Overcomplete Latent Variable Models through Tensor Methods

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    We provide guarantees for learning latent variable models emphasizing on the overcomplete regime, where the dimensionality of the latent space can exceed the observed dimensionality. In particular, we consider multiview mixtures, spherical Gaussian mixtures, ICA, and sparse coding models. We provide tight concentration bounds for empirical moments through novel covering arguments. We analyze parameter recovery through a simple tensor power update algorithm. In the semi-supervised setting, we exploit the label or prior information to get a rough estimate of the model parameters, and then refine it using the tensor method on unlabeled samples. We establish that learning is possible when the number of components scales as k=o(dp/2)k=o(d^{p/2}), where dd is the observed dimension, and pp is the order of the observed moment employed in the tensor method. Our concentration bound analysis also leads to minimax sample complexity for semi-supervised learning of spherical Gaussian mixtures. In the unsupervised setting, we use a simple initialization algorithm based on SVD of the tensor slices, and provide guarantees under the stricter condition that k≤βdk\le \beta d (where constant β\beta can be larger than 11), where the tensor method recovers the components under a polynomial running time (and exponential in β\beta). Our analysis establishes that a wide range of overcomplete latent variable models can be learned efficiently with low computational and sample complexity through tensor decomposition methods.Comment: Title change

    Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

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    The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

    Recovery from Linear Measurements with Complexity-Matching Universal Signal Estimation

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    We study the compressed sensing (CS) signal estimation problem where an input signal is measured via a linear matrix multiplication under additive noise. While this setup usually assumes sparsity or compressibility in the input signal during recovery, the signal structure that can be leveraged is often not known a priori. In this paper, we consider universal CS recovery, where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself. Inspired by Kolmogorov complexity and minimum description length, we focus on a maximum a posteriori (MAP) estimation framework that leverages universal priors to match the complexity of the source. Our framework can also be applied to general linear inverse problems where more measurements than in CS might be needed. We provide theoretical results that support the algorithmic feasibility of universal MAP estimation using a Markov chain Monte Carlo implementation, which is computationally challenging. We incorporate some techniques to accelerate the algorithm while providing comparable and in many cases better reconstruction quality than existing algorithms. Experimental results show the promise of universality in CS, particularly for low-complexity sources that do not exhibit standard sparsity or compressibility.Comment: 29 pages, 8 figure
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