The 1-bit compressed sensing framework enables the recovery of a sparse vector x from the sign information of each entry of its linear transformation. Discarding the amplitude information can significantly reduce the ...
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The 1-bit compressed sensing framework enables the recovery of a sparse vector x from the sign information of each entry of its linear transformation. Discarding the amplitude information can significantly reduce the amount of data, which is highly beneficial in practical applications. In this paper, we present a Bayesian approach to signal reconstruction for 1-bit compressed sensing and analyze its typical performance using statistical mechanics. As a basic setup, we consider the case that the measuring matrix Phi has i.i.d entries and the measurements y are noiseless. Utilizing the replica method, we show that the Bayesian approach enables better reconstruction than the l(1)-norm minimization approach, asymptotically saturating the performance obtained when the non-zero entry positions of the signal are known, for signals whose non-zero entries follow zero mean Gaussian distributions. We also test a messagepassing algorithm for signal reconstruction on the basis of belief propagation. The results of numerical experiments are consistent with those of the theoretical analysis.
The distributed compressed sensing framework provides an efficient compression scheme of multichannel signals that are sparse in some domains and highly correlated with one another. In particular, a signal model calle...
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The distributed compressed sensing framework provides an efficient compression scheme of multichannel signals that are sparse in some domains and highly correlated with one another. In particular, a signal model called the joint sparse model 2 (JSM-2) or multiple measurement vector problem, in which all sparse signals share their support, is important for dealing with practical problems such as magnetic resonance imaging and magnetoencephalography. In this paper, we investigate the typical reconstruction performance of JSM-2 problems for two schemes. One is l(2,1)-norm minimization reconstruction and the other is Bayesian optimal reconstruction. Employing the replica method, we show that the reconstruction performance of both schemes which exploit the knowledge of the sharing of the signal support overcomes that of their corresponding approaches for the single-channel compressed sensing problem. We also develop a computationally feasible approximate algorithm for performing the Bayes optimal scheme to validate our theoretical estimation. Our replica-based analysis numerically indicates that the spinodal point of the Bayesian reconstruction disappears, which implies that a fundamental reconstruction limit can be achieved by the BP-based approximate algorithm in a practical amount of time when the number of channels is sufficiently large. The results of the numerical experiments of both reconstruction schemes agree excellently with the theoretical evaluation.
We investigate leave-one-out cross validation (CV) as a determinator of the weight of the penalty term in the least absolute shrinkage and selection operator (LASSO). First, on the basis of the messagepassing algorit...
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We investigate leave-one-out cross validation (CV) as a determinator of the weight of the penalty term in the least absolute shrinkage and selection operator (LASSO). First, on the basis of the messagepassing algorithm and a perturbative discussion assuming that the number of observations is sufficiently large, we provide simple formulas for approximately assessing two types of CV errors, which enable us to significantly reduce the necessary cost of computation. These formulas also provide a simple connection of the CV errors to the residual sums of squares between the reconstructed and the given measurements. Second, on the basis of this finding, we analytically evaluate the CV errors when the design matrix is given as a simple random matrix in the large size limit by using the replica method. Finally, these results are compared with those of numerical simulations on finite-size systems and are confirmed to be correct. We also apply the simple formulas of the first type of CV error to an actual dataset of the supernovae.
This work considers the compressed sensing (CS) of i.i.d. signals with sparse measurement matrices and belief-propagation (BP) reconstruction. In general, BP reconstruction for CS requires the passing of messages that...
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ISBN:
(纸本)9781509018253
This work considers the compressed sensing (CS) of i.i.d. signals with sparse measurement matrices and belief-propagation (BP) reconstruction. In general, BP reconstruction for CS requires the passing of messages that are distributions over the real numbers. To implement this in practice, one typically uses either quantized distributions or a Gaussian approximation. In this work, we use density evolution to compare the reconstruction performance of these two methods. Since the reconstruction performance depends on the signal realization, this analysis makes use of a novel change of variables to analyze the performance for a typical signal. Simulation results are provided to support the results.
作者：
Barre, J.Univ Nice
Lab JA Dieudonne CNRS UMR 6621 F-06108 Nice 2 France
We address the problem of retrieving information from a noisy version of the ' knowledge networks' introduced by Maslov and Zhang ( 2001 Phys. Rev. Lett. 87 248701). We map this problem onto a disordered stati...
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We address the problem of retrieving information from a noisy version of the ' knowledge networks' introduced by Maslov and Zhang ( 2001 Phys. Rev. Lett. 87 248701). We map this problem onto a disordered statistical mechanics model, which opens the door to many analytical and numerical approaches. We give the replica symmetric solution, compare it with numerical simulations, and. finally discuss an application to real data from the United States Senate.
We study hard constraint satisfaction problems with a decimation approach based on messagepassingalgorithms. Decimation induces a renormalization flow in the space of problems, and we exploit the fact that this flow...
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We study hard constraint satisfaction problems with a decimation approach based on messagepassingalgorithms. Decimation induces a renormalization flow in the space of problems, and we exploit the fact that this flow transforms some of the constraints into linear constraints over GF(2). In particular, when the flow hits the subspace of linear problems, one can stop the decimation and use Gaussian elimination. We introduce a new decimation algorithm which uses this linear structure and shows a strongly improved performance with respect to the usual decimation methods for some of the hardest locked occupation problems.
We propose a new method for obtaining hierarchical clustering based on the optimization of a cost function over trees of limited depth, and we derive a message-passing method that allows one to use it efficiently. The...
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We propose a new method for obtaining hierarchical clustering based on the optimization of a cost function over trees of limited depth, and we derive a message-passing method that allows one to use it efficiently. The method and the associated algorithm can be interpreted as a natural interpolation between two well-known approaches, namely that of single linkage and the recently presented affinity propagation. We analyse using this general scheme three biological/medical structured data sets (human population based on genetic information, proteins based on sequences and verbal autopsies) and show that the interpolation technique provides new insight.
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