Sample complexity for 1-bit compressed sensing and sparse classification

Gupta, Ankit; Nowak, Robert; Recht, Benjamin

doi:10.1109/isit.2010.5513510

Cited by 70 publications

(79 citation statements)

References 12 publications

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“…The results established here also address a previously open question posed in [23]. That work considered the problem of support recovery from measurements corrupted by additive Gaussian noise and quantized to a single bit and proposed a procedure based on non-adaptive measurements requiring O(Dk log n) measurements for exact recovery, where D := maxi,j∈S |xi|/|xj| is the dynamic range of the signal.…”

Section: Our Contributionssupporting

confidence: 57%

“…For example, constraints on measurement precision (which could be due to acquisition hardware limitations) may be modeled as measurement quantization. Several existing works have examined the general effects of quantization in CS [16]- [20], some focusing exclusively on the case where the measurements are highly quantized to a single bit [21]- [23]. Other nonGaussian measurement uncertainty models may include corruption by a moderate number of large-valued "outliers."…”

Section: A Motivationmentioning

confidence: 99%

“…Indeed, the majority of the work examining CS under non-Gaussian corruption models has focused primarily on estimation of the unknown signal x, treating the non-Gaussian uncertainty as noise [16], explicitly accounting for it with additional sampling or algorithmic constraints [18]- [22], [24]- [27], or explicitly designing it in order to yield optimal estimation error performance using a particular recovery technique [17]. One exception is the work in [23], which considered the problem of support recovery from measurements corrupted by Gaussian noise and subsequently quantized to a single bit.…”

Section: A Motivationmentioning

confidence: 99%

See 2 more Smart Citations

Robust support recovery using sparse compressive sensing matrices

Haupt

Baraniuk

2011

2011 45th Annual Conference on Information Sciences and Systems

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Abstract-This paper considers the task of recovering the support of a sparse, high-dimensional vector from a small number of measurements. The procedure proposed here, which we call the Sign-Sketch procedure, is shown to be a robust recovery method in settings where the measurements are corrupted by various forms of uncertainty, including additive Gaussian noise and (possibly unbounded) outliers, and even subsequent quantization of the measurements to a single bit. The Sign-Sketch procedure employs sparse random measurement matrices, and utilizes a computationally efficient support recovery procedure that is a variation of a technique from the sketching literature. We show here that O(max {k log(n − k), k log k}) non-adaptive linear measurements suffice to recover the support of any unknown n-dimensional vector having no more than k nonzero entries, and that our proposed procedure requires at most O(n log n) total operations for both acquisition and inference.

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Section: Our Contributionssupporting

confidence: 57%

Section: A Motivationmentioning

confidence: 99%

Section: A Motivationmentioning

confidence: 99%

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Robust support recovery using sparse compressive sensing matrices

Haupt

Baraniuk

2011

2011 45th Annual Conference on Information Sciences and Systems

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“…The difficulty is that the first two of these constraints are non-convex, and thus the only known program which is known to return such an estimate is 0 minimization with the unit norm constraint-this is generally considered to be intractable. Gupta et al [16] demonstrate that one may tractably recover the support of x from O(s log n) measurements. They give two measurement schemes.…”

Section: Introductionmentioning

confidence: 99%

One‐Bit Compressed Sensing by Linear Programming

Plan

Vershynin

2013

Comm Pure Appl Math

333

318

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Abstract. We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x ∈ R n from the signs of O(s log 2 (n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellations.

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“…In CS, the sparsity leads to the minimization of 1 norm of the unknown signal in the recovery algorithm, because A more related CS problem to the CL problem is 1-bit compressed sensing (Gupta et al 2010). Different from CS which recovers the sparse signal from a few of its random projections, 1-bit CS aims to recover the support set of the sparse signal x from a few of its random projection signs y = sign(xA).…”

Section: Cs and CL Develop Different Recovery Algorithms By Exploitinmentioning

confidence: 99%

Compressed labeling on distilled labelsets for multi-label learning

Zhou

Tao

2012

Mach Learn

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Directly applying single-label classification methods to the multi-label learning problems substantially limits both the performance and speed due to the imbalance, dependence and high dimensionality of the given label matrix. Existing methods either ignore these three problems or reduce one with the price of aggravating another. In this paper, we propose a {0, 1} label matrix compression and recovery method termed "compressed labeling (CL)" to simultaneously solve or at least reduce these three problems. CL first compresses the original label matrix to improve balance and independence by preserving the signs of its Gaussian random projections. Afterward, we directly utilize popular binary classification methods (e.g., support vector machines) for each new label. A fast recovery algorithm is developed to recover the original labels from the predicted new labels. In the recovery algorithm, a "labelset distilling method" is designed to extract distilled labelsets (DLs), i.e., the frequently appeared label subsets from the original labels via recursive clustering and subtraction. Given a distilled and an original label vector, we discover that the signs of their random projections have an explicit joint distribution that can be quickly computed from a geometric inference. Based on this observation, the original label vector is exactly determined after performing a series of Kullback-Leibler divergence based hypothesis tests on the distribution about the new labels. CL significantly improves the balance Mach Learn (2012) 88:69-126 of the training samples and reduces the dependence between different labels. Moreover, it accelerates the learning process by training fewer binary classifiers for compressed labels, and makes use of label dependence via DLs based tests. Theoretically, we prove the recovery bounds of CL which verifies the effectiveness of CL for label compression and multi-label classification performance improvement brought by label correlations preserved in DLs. We show the effectiveness, efficiency and robustness of CL via 5 groups of experiments on 21 datasets from text classification, image annotation, scene classification, music categorization, genomics and web page classification.

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Sample complexity for 1-bit compressed sensing and sparse classification

Cited by 70 publications

References 12 publications

Robust support recovery using sparse compressive sensing matrices

Robust support recovery using sparse compressive sensing matrices

One‐Bit Compressed Sensing by Linear Programming

Compressed labeling on distilled labelsets for multi-label learning

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