Near-Optimal Bounds for Binary Embeddings of Arbitrary Sets

Oymak, Samet; Recht, Ben

doi:10.48550/arxiv.1512.04433

Cited by 12 publications

(31 citation statements)

References 20 publications

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“…One of key ingredients of our proof of the RAIC is the local binary embedding (local sensitivity hashing) property by Oymak and Recht [26]. Bilyk and Lacey [29] also reported a similar property.…”

Section: E Local Binary Embedding For Small Regions At Xmentioning

confidence: 81%

“…• Local Binary Embedding (LBE) [26]: The LBE is a refined version of the Binary ε-Stable Embedding (BεSE) by Jacques et.al. [15].…”

Section: B Sketch Of the Proof Of The Raicmentioning

confidence: 99%

“…Theorem 16 (Local δ -binary embedding [26], [29]). Let A ∈ R m×N be a standard Gaussian random matrix.…”

Section: E Local Binary Embedding For Small Regions At Xmentioning

confidence: 99%

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NBIHT: An Efficient Algorithm for 1-bit Compressed Sensing with Optimal Error Decay Rate

Friedlander¹,

Jeong²,

Plan³

et al. 2020

Preprint

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The Binary Iterative Hard Thresholding (BIHT) algorithm is a popular reconstruction method for one-bit compressed sensing due to its simplicity and fast empirical convergence. There have been several works about BIHT but a theoretical understanding of the corresponding approximation error and convergence rate still remains open.This paper shows that the normalized version of BIHT (NBHIT) achieves an approximation error rate optimal up to logarithmic factors. More precisely, using m one-bit measurements of an s-sparse vector x, we prove that the approximation error of NBIHT is of order O 1 m up to logarithmic factors, which matches the information-theoretic lower bound Ω 1 m proved by Jacques, Laska, Boufounos, and Baraniuk in 2013. To our knowledge, this is the first theoretical analysis of a BIHT-type algorithm that explains the optimal rate of error decay empirically observed in the literature. This also makes NBIHT the first provable computationally-efficient one-bit compressed sensing algorithm that breaks the inverse square root error decay rate O 1 m 1/2 .

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Section: E Local Binary Embedding For Small Regions At Xmentioning

confidence: 81%

“…• Local Binary Embedding (LBE) [26]: The LBE is a refined version of the Binary ε-Stable Embedding (BεSE) by Jacques et.al. [15].…”

Section: B Sketch Of the Proof Of The Raicmentioning

confidence: 99%

See 1 more Smart Citation

NBIHT: An Efficient Algorithm for 1-bit Compressed Sensing with Optimal Error Decay Rate

Friedlander¹,

Jeong²,

Plan³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In this specific situation, it turns out that Assumption 4.1 is very compatible with uniform bounds for binary embeddings, which allows us to make use of related results from the literature. Our first application is based on the following embedding guarantee by Oymak and Recht [OR15] for noiseless, Gaussian 1-bit measurements (cf. Subsection 3.1).…”

Section: Uniform Recovery Without Increment Conditionsmentioning

confidence: 99%

A Unified Approach to Uniform Signal Recovery From Non-Linear Observations

Genzel¹,

Stollenwerk²

2020

Preprint

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Recent advances in quantized compressed sensing and high-dimensional estimation have shown that signal recovery is even feasible under strong non-linear distortions in the observation process. An important characteristic of associated guarantees is uniformity, i.e., recovery succeeds for an entire class of structured signals with a fixed measurement ensemble. However, despite significant results in various special cases, a general understanding of uniform recovery from non-linear observations is still missing. This paper develops a unified approach to this problem under the assumption i.i.d. sub-Gaussian measurement vectors. Our main result shows that a simple least-squares estimator with any convex constraint can serve as a universal recovery strategy, which is outlier robust and does not require explicit knowledge of the underlying non-linearity. Based on empirical process theory, a key technical novelty is an approximative increment condition that can be implemented for all common types of non-linear models. This flexibility allows us to apply our approach to a variety of problems in quantized compressed sensing and high-dimensional statistics, leading to several new and improved guarantees. Each of these applications is accompanied by a conceptually simple and systematic proof, which does not rely on any deeper properties of the observation model. On the other hand, known local stability properties can be incorporated into our framework in a plug-and-play manner, thereby implying near-optimal error bounds.

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“…The theoretical results for fast binary embedding techniques are rather limited [13,22,23]. Related to us, very recently Yu et al provided an analysis of circulant projections.…”

Section: Introductionmentioning

confidence: 99%

Near-optimal sample complexity bounds for circulant binary embedding

Oymak

Thrampoulidis

Hassibi

2017

2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Binary embedding is the problem of mapping points from a high-dimensional space to a Hamming cube in lower dimension while preserving pairwise distances. An efficient way to accomplish this is to make use of fast embedding techniques involving Fourier transform e.g. circulant matrices. While binary embedding has been studied extensively, theoretical results on fast binary embedding are rather limited. In this work, we build upon the recent literature to obtain significantly better dependencies on the problem parameters. A set of N points in R n can be properly embedded into the Hamming cube {±1} k with δ distortion, by using k ∼ δ −3 log N samples which is optimal in the number of points N and compares well with the optimal distortion dependency δ −2 . Our optimal embedding result applies in the regime log N ≲ n 1 3 . Furthermore, if the looser condition log N ≲ √ n holds, we show that all but an arbitrarily small fraction of the points can be optimally embedded. We believe our techniques can be useful to obtain improved guarantees for other nonlinear embedding problems.

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Near-Optimal Bounds for Binary Embeddings of Arbitrary Sets

Cited by 12 publications

References 20 publications

NBIHT: An Efficient Algorithm for 1-bit Compressed Sensing with Optimal Error Decay Rate

NBIHT: An Efficient Algorithm for 1-bit Compressed Sensing with Optimal Error Decay Rate

A Unified Approach to Uniform Signal Recovery From Non-Linear Observations

Near-optimal sample complexity bounds for circulant binary embedding

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