8,495 research outputs found

    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

    Adaptive Compressed Sensing for Support Recovery of Structured Sparse Sets

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    This paper investigates the problem of recovering the support of structured signals via adaptive compressive sensing. We examine several classes of structured support sets, and characterize the fundamental limits of accurately recovering such sets through compressive measurements, while simultaneously providing adaptive support recovery protocols that perform near optimally for these classes. We show that by adaptively designing the sensing matrix we can attain significant performance gains over non-adaptive protocols. These gains arise from the fact that adaptive sensing can: (i) better mitigate the effects of noise, and (ii) better capitalize on the structure of the support sets.Comment: to appear in IEEE Transactions on Information Theor

    One-bit compressive sensing with norm estimation

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    Consider the recovery of an unknown signal x{x} from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that x{x} is sparse, and that the measurements are of the form sign(ai,x){±1}\operatorname{sign}(\langle {a}_i, {x} \rangle) \in \{\pm1\}. Since such measurements give no information on the norm of x{x}, recovery methods from such measurements typically assume that x2=1\| {x} \|_2=1. We show that if one allows more generally for quantized affine measurements of the form sign(ai,x+bi)\operatorname{sign}(\langle {a}_i, {x} \rangle + b_i), and if the vectors ai{a}_i are random, an appropriate choice of the affine shifts bib_i allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed x{x} in the annulus rx2Rr \leq \| {x} \|_2 \leq R, one may estimate the norm x2\| {x} \|_2 up to additive error δ\delta from mR4r2δ2m \gtrsim R^4 r^{-2} \delta^{-2} such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors x{x} within a Euclidean ball of known radius.Comment: 20 pages, 2 figure
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