Fast binary embeddings, and quantized compressed sensing with structured matrices

Huynh, Thang; Saab, Rayan

doi:10.48550/arxiv.1801.08639

Cited by 6 publications

(20 citation statements)

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“…Related works: Reconstruction of low-complexity signals from QCS observations has been studied in the context of 1-bit CS [24,13,25,15] and multi-bit quantization [26,27,8]. Most of these works focus on estimating such signals from their quantized or non-linearly disturbed sub-Gaussian random projections.…”

Section: Contributionsmentioning

confidence: 99%

“…Most of these works focus on estimating such signals from their quantized or non-linearly disturbed sub-Gaussian random projections. The studies [8] and [15] are two exceptions that use, respectively, BOS and PCE constructions, and subsampled Gaussian random circulant sensing matrix. However, both works are restricted to sparse signal estimations.…”

Section: Contributionsmentioning

confidence: 99%

“…up to a vectorization. Equivalently, saturating the condition on m in (8) and inverting this relation with respect to provides = O((W(K )/m) 1/2 ), up to missing log factors. We can finally conclude this section and say that, within the precise context described above, uniformly or non-uniformly over the generation of ξ, PBP provides, w.h.p., sparse signal or low-rank matrix estimates whose reconstruction error decays like…”

Section: Error Decay Analysis Of Pbpmentioning

confidence: 99%

“…Therefore, a more realistic Quantized CS (QCS) model lies in estimating a low-complexity signal x ∈ K from y = Q g (Φx), where Q g : u ∈ R m → Q g (u) ∈ A ⊂ R m is a general quantization function (or quantizer ) mapping m-dimensional vectors to vectors in a discrete set (or codebook ) A. Many quantizers have been studied in QCS, e.g., Σ∆quantization [7,8], non-regular scalar quantizers [9], non-regular binned quantization [10,11], and vector quantization by frame permutation [12]. They aim at easing the impact of the quantizer on the estimation of x from y, by some well-designed algorithms achieving fast (e.g., polynomial or exponential) reconstruction error decay when m increases [7,13,14].…”

Section: Introductionmentioning

confidence: 99%

“…These works are mostly dominated by the use of sub-Gaussian random matrices. Only two recent studies escape from this domination: [15] uses partial circulant ensembles with Gaussian random entries in 1-bit CS, and [8] leverages fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems (BOS) and partial circulant ensembles (PCE) with noise-shaping quantization (e.g., Σ∆).…”

Section: Introductionmentioning

confidence: 99%

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Taking the Edge off Quantization: Projected Back Projection in Dithered Compressive Sensing

Xu¹,

Schellekens²,

Jacques³

2018

2018 IEEE Statistical Signal Processing Workshop (SSP)

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Quantized compressive sensing (QCS) deals with the problem of representing compressive signal measurements with finite precision representation, i.e., a mandatory process in any practical sensor design. To characterize the signal reconstruction quality in this framework, most of the existing theoretical analyses lie heavily on the quantization of sub-Gaussian random projections (e.g., Gaussian or Bernoulli). We show here that a simple uniform scalar quantizer is compatible with a large class of random sensing matrices known to respect, with high probability, the restricted isometry property (RIP). Critically, this compatibility arises from the addition of a uniform random vector, or dithering, to the linear signal observations before quantization. In this setting, we prove the existence of (at least) one signal reconstruction method, i.e., the projected back projection (PBP), whose reconstruction error decays when the number of quantized measurements increases. This holds with high probability in the estimation of sparse signals and low-rank matrices. We validate numerically the predicted error decay as the number of measurements increases.

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Section: Contributionsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Error Decay Analysis Of Pbpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Taking the Edge off Quantization: Projected Back Projection in Dithered Compressive Sensing

Xu¹,

Schellekens²,

Jacques³

2018

2018 IEEE Statistical Signal Processing Workshop (SSP)

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Quantized Compressed Sensing: A Survey

Dirksen

2019

Applied and Numerical Harmonic Analysis

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The field of quantized compressed sensing investigates how to jointly design a measurement matrix, quantizer, and reconstruction algorithm in order to accurately reconstruct low-complexity signals from a minimal number of measurements that are quantized to a finite number of bits. In this short survey, we give an overview of the state-of-the-art rigorous reconstruction results that have been obtained for three popular quantization models: one-bit quantization, uniform scalar quantization, and noise-shaping methods.

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One-bit compressed sensing with partial Gaussian circulant matrices

Dirksen

Jung

Rauhut

2019

Information and Inference: A Journal of the IMA

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We present optimal sample complexity estimates for one-bit compressed sensing problems in a realistic scenario: the procedure uses a structured matrix (a randomly sub-sampled circulant matrix) and is robust to analog pre-quantization noise as well as to adversarial bit corruptions in the quantization process. Our results imply that quantization is not a statistically expensive procedure in the presence of nontrivial analog noise: recovery requires the same sample size one would have needed had the measurement matrix been Gaussian and the noisy analog measurements been given as data.

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Fast binary embeddings, and quantized compressed sensing with structured matrices

Cited by 6 publications

References 0 publications

Taking the Edge off Quantization: Projected Back Projection in Dithered Compressive Sensing

Taking the Edge off Quantization: Projected Back Projection in Dithered Compressive Sensing

Quantized Compressed Sensing: A Survey

One-bit compressed sensing with partial Gaussian circulant matrices

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