Fast Binary Embeddings with Gaussian Circulant Matrices: Improved Bounds

Dirksen, Sjoerd; Stollenwerk, Alexander

doi:10.1007/s00454-017-9964-x

Cited by 10 publications

(11 citation statements)

References 13 publications

(22 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The setup and results are related to ours, but bear several important differences, including (i) the use of regularization; (ii) the focus on sub-Gaussian noise added before quantization; (iii) the use of dithering-type thresholds in the measurements; (iv) the consideration of sparse signals instead of generative priors; and (v) the use of a random (rather than fixed) index set in the partial circulant matrix. Some theoretical guarantees for circulant binary embeddings, which are closely related to 1-bit CS with partial Gaussian circulant matrices, have been presented in [24], [33]- [35]. On the other hand, based on successful applications of deep generative models, instead of assuming the signal vector is sparse, it has been recently popular to assume that the signal vector lies in the range of a generative model in the problem of CS [3], [36]- [43].…”

Section: A Related Workmentioning

confidence: 99%

Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

Liu¹,

Ghosh²,

Han³

et al. 2021

Preprint

View full text Add to dashboard Cite

In 1-bit compressive sensing, each measurement is quantized to a single bit, namely the sign of a linear function of an unknown vector, and the goal is to accurately recover the vector. While it is most popular to assume a standard Gaussian sensing matrix for 1-bit compressive sensing, using structured sensing matrices such as partial Gaussian circulant matrices is of significant practical importance due to their faster matrix operations. In this paper, we provide recovery guarantees for a correlation-based optimization algorithm for robust 1-bit compressive sensing with randomly signed partial Gaussian circulant matrices and generative models. Under suitable assumptions, we match guarantees that were previously only known to hold for i.i.d. Gaussian matrices that require significantly more computation. We make use of a practical iterative algorithm, and perform numerical experiments on image datasets to corroborate our theoretical results.

show abstract

Section: A Related Workmentioning

confidence: 99%

Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

Liu¹,

Ghosh²,

Han³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Unlike previous works on quantization under frame or Compressed Sensing measurements (see, e.g., [36,6,28,5,23,29,3,30,21,2,19,27,34,35,25,18,19,20]), where samples are assumed to be taken by random Gaussian/sub-Gaussian or Fourier measurements, here we allow a direct quantization on each pixel and therefore ensure the maximal practicality.…”

Section: Contributionmentioning

confidence: 99%

Sigma Delta quantization for images

Lyu¹,

Wang²

2020

Preprint

View full text Add to dashboard Cite

In this paper, we propose the first adaptive quantization method for direct digital image acquisition that yields a better information conversion rate than the state-of-theart method in cameras. This new method allows a rich color-palette to be reconstructed by using extremely low bits for each pixel and therefore is beneficial for capturing scenes with high-contrast. The work is motivated by recent results on super-resolution for sparse signals, but is free from the usual separation requirement between spikes. We assume natural images are of small TV norms of order 1 or 2, and design an appropriate reconstruction algorithm that reduces the reconstruction error from the known O(where P is the number of pixels and s is the number of edges in the image. Our numerical experiments confirm these theoretical findings and further show that a dramatic increase in the photo quality can be achieved when the photo is taken in a dark environment.

show abstract

“…To address the first point above, researchers have tried to design other binary embeddings which can be implemented more efficiently. For example, such embeddings include circulant binary embedding [72,53,24], structured hashed projections [16], binary embeddings based on Walsh-Hadamard matrices and partial Gaussian Toeplitz matrices with random column flips [71]. To our knowledge, all of these results do not address the second point above; they use the sign function (1.3) (which is an instance of Memoryless Scalar Quantization) to quantize the measurements.…”

Section: Drawback (Ii)mentioning

confidence: 99%

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Huynh

Saab

2019

Comm Pure Appl Math

View full text Add to dashboard Cite

This paper deals with two related problems, namely distance‐preserving binary embeddings and quantization for compressed sensing. First, we propose fast methods to replace points from a subset Χ ⊂ ℝn, associated with the euclidean metric, with points in the cube {±1}m, and we associate the cube with a pseudometric that approximates euclidean distance among points in Χ. Our methods rely on quantizing fast Johnson‐Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise shaping, and include sigma‐delta schemes and distributed noise‐shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in m, depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson‐Lindenstrauss maps while preserving ℓ2 norms. Second, we again consider noise‐shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for sigma‐delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise‐shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems. © 2019 Wiley Periodicals, Inc.

show abstract

Fast Binary Embeddings with Gaussian Circulant Matrices: Improved Bounds

Cited by 10 publications

References 13 publications

Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

Sigma Delta quantization for images

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Contact Info

Product

Resources

About