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45
Multistage Adaptive Estimation of Sparse Signals
, 2012
"... This paper considers sequential adaptive estimation of sparse signals under a constraint on the total sensing effort. A dynamic programming formulation is derived for the allocation of sensing resources to minimize a cost function related to mean squared estimation error. Allocation policies are dev ..."
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Cited by 19 (6 self)
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This paper considers sequential adaptive estimation of sparse signals under a constraint on the total sensing effort. A dynamic programming formulation is derived for the allocation of sensing resources to minimize a cost function related to mean squared estimation error. Allocation policies are developed based on the method of openloop feedback control. These policies are optimal in the twostage case and improve monotonically thereafter with the number of stages. Numerical simulations show gains up to several dB as compared to recently proposed adaptive methods, and dramatic gains approaching the oracle limit as compared to nonadaptive estimation.
NearOptimal Adaptive Compressed Sensing
, 1306
"... Abstract—This paper proposes a simple adaptive sensing and group testing algorithm for sparse signal recovery. The algorithm, termed Compressive Adaptive Sense and Search (CASS), is shown to be nearoptimal in that it succeeds at the lowest possible signaltonoiseratio (SNR) levels. Like tradition ..."
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Cited by 19 (0 self)
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Abstract—This paper proposes a simple adaptive sensing and group testing algorithm for sparse signal recovery. The algorithm, termed Compressive Adaptive Sense and Search (CASS), is shown to be nearoptimal in that it succeeds at the lowest possible signaltonoiseratio (SNR) levels. Like traditional compressed sensing based on random nonadaptive design matrices, the CASS algorithm requires only k log n measurements to recover a ksparse signal of dimension n. However,CASSsucceedsatSNR levels that are a factor log n less than required by standard compressed sensing. From the point of view of constructing and implementing the sensing operation as well as computing the reconstruction, the proposed algorithm is substantially less computationally intensive than standard compressed sensing. CASS is also demonstrated to perform considerably better in practice through simulation. To the best of our knowledge, this is the first demonstration of an adaptive compressed sensing algorithm with nearoptimal theoretical guarantees and excellent practical performance. This paper also shows that methods like compressed sensing, group testing, and pooling have an advantage beyond simply reducing the number of measurements or tests – adaptive versions of such methods can also improve detection and estimation performance when compared to nonadaptive direct (uncompressed) sensing. I.
Detection boundary in sparse regression
, 2010
"... We study the problem of detection of a pdimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimens ..."
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Cited by 18 (6 self)
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We study the problem of detection of a pdimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to the infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the highdimensional setting (p ≫ n). The main message is that, under some conditions, the detection boundary phenomenon that has been proved for the Gaussian sequence model, extends to highdimensional linear regression. Finally, we establish the detection boundaries when the variance of the noise is unknown. Interestingly, the detection boundaries sometimes depend on the knowledge of the variance in a highdimensional setting.
Sequentially Designed Compressed Sensing
"... A sequential adaptive compressed sensing procedure for signal support recovery is proposed and analyzed. The procedure is based on the principle of distilled sensing, and makes used of sparse sensing matrices to perform sketching observations able to quickly identify irrelevant signal components. I ..."
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Cited by 18 (5 self)
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A sequential adaptive compressed sensing procedure for signal support recovery is proposed and analyzed. The procedure is based on the principle of distilled sensing, and makes used of sparse sensing matrices to perform sketching observations able to quickly identify irrelevant signal components. It is shown that adaptive compressed sensing enables recovery of weaker sparse signals than those that can be recovered using traditional nonadaptive compressed sensing approaches.
Capturing ridge functions in high dimensions from point queries
, 2010
"... Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on R N with smoothness of order s can in general be captured with accuracy at most O(n −s/N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s ..."
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Cited by 13 (0 self)
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Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on R N with smoothness of order s can in general be captured with accuracy at most O(n −s/N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x) = g(a · x) when both a ∈ R N and g ∈ C[0, 1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g ∈ C s [0, 1]. We also study the role of sparsity or compressibility of a in such query problems.
Adaptive sensing performance lower bounds for sparse signal estimation and testing,” arXiv preprint arXiv:1206.0648
, 2012
"... Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peerreview. There can be important differences between the submitted version and the official published version of record. People interested in the research are ..."
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Cited by 13 (3 self)
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Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peerreview. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profitmaking activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim. In memory of Yuri Ingster This paper gives a precise characterization of the fundamental limits of adaptive sensing for diverse estimation and testing problems concerning sparse signals. We consider in particular the setting introduced in (IEEE Trans. Inform. Theory 57 (2011) 62226235) and show necessary conditions on the minimum signal magnitude for both detection and estimation: if x ∈ R n is a sparse vector with s nonzero components then it can be reliably detected in noise provided the magnitude of the nonzero components exceeds √ 2/s. Furthermore, the signal support can be exactly identified provided the minimum magnitude exceeds √ 2 log s. Notably there is no dependence on n, the extrinsic signal dimension. These results show that the adaptive sensing methodologies proposed previously in the literature are essentially optimal, and cannot be substantially improved. In addition, these results provide further insights on the limits of adaptive compressive sensing.
Recovering GraphStructured Activations using Adaptive Compressive Measurements
"... Abstract—We study the localization of a cluster of activated vertices in a graph, from adaptively designed compressive measurements. We propose a hierarchical partitioning of the graph that groups the activated vertices into few partitions, so that a topdown sensing procedure can identify these par ..."
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Cited by 8 (1 self)
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Abstract—We study the localization of a cluster of activated vertices in a graph, from adaptively designed compressive measurements. We propose a hierarchical partitioning of the graph that groups the activated vertices into few partitions, so that a topdown sensing procedure can identify these partitions, and hence the activations, using few measurements. By exploiting the cluster structure, we are able to provide localization guarantees at weaker signal to noise ratios than in the unstructured setting. We complement this performance guarantee with an information theoretic lower bound, providing a necessary signaltonoise ratio for any algorithm to successfully localize the cluster. We verify our analysis with some simulations, demonstrating the practicality of our algorithm. I.
On the power of adaptivity in matrix completion and approximation
, 2014
"... We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sam ..."
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Cited by 4 (3 self)
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We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. For exact recovery of a lowrank matrix, our algorithm judiciously selects a few columns to observe in full and, with few additional measurements, projects the remaining columns onto their span. This algorithm exactly recovers an n × n rank r matrix using O(nrµ0 log2(r)) observations, where µ0 is a coherence parameter on the column space of the matrix. In addition to completely eliminating any row space assumptions that have pervaded the literature, this algorithm enjoys a better sample complexity than any existing matrix completion algorithm. To certify that this improvement is due to adaptive sampling, we establish that row space coherence is necessary for passive sampling algorithms to achieve nontrivial sample complexity bounds. For constructing a lowrank approximation to a highrank input matrix, we propose a simple algorithm that thresholds the singular values of a zerofilled version of the input matrix. The algorithm computes an approximation that is nearly as good as the best rankr approximation using O(nrµ log2(n)) samples, where µ is a slightly different coherence parameter on the matrix columns. Again we eliminate assumptions on the row space. 1
The capacity of adaptive group testing
 In 2013 IEEE International Symposium on Information Theory, Istanbul Turkey
, 2013
"... Abstract—We define capacity for group testing problems and deduce bounds for the capacity of a variety of noisy models, based on the capacity of equivalent noisy communication channels. For noiseless adaptive group testing we prove an informationtheoretic lower bound which tightens a bound of Chan ..."
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Cited by 4 (3 self)
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Abstract—We define capacity for group testing problems and deduce bounds for the capacity of a variety of noisy models, based on the capacity of equivalent noisy communication channels. For noiseless adaptive group testing we prove an informationtheoretic lower bound which tightens a bound of Chan et al. This can be combined with a performance analysis of a version of Hwang’s adaptive group testing algorithm, in order to deduce the capacity of noiseless and erasure group testing models. I. INTRODUCTION AND NOTATION We consider the problem of noiseless group testing, as introduced by Dorfman [1] in the context of testing populations of soldiers for syphilis. Group testing has recently been used to screen DNA samples for rare alleles by a pooling strategy,