Noise-Shaping Quantization Methods for Frame-Based and Compressive Sampling Systems

Chou, Evan; Güntürk, C. Sınan; Krahmer, Felix; Saab, Rayan; Yılmaz, Özgür

doi:10.1007/978-3-319-19749-4_4

Cited by 29 publications

(33 citation statements)

References 41 publications

(116 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We show that noise-shaping quantization approaches, namely Σ∆ quantization (e.g., [32,42,20]) and distributed noise-shaping quantization [17] significantly outperform the naive (but commonly used) quantization approach, memoryless scalar quantization. They yield polynomial and exponential error decay, respectively, as a function of the number of measurements.…”

Section: Contributions Related To the Quantization Of Compressed Sensmentioning

confidence: 99%

“…While one can in principle still get a reduction in the error by using a finer quantization alphabet (i.e., a smaller δ), this is often not feasible in practice because the quantization alphabet is fixed once the hardware is built, or not desirable as one may prefer a simple, e.g., 1-bit embedding. In order to address this problem, noise-shaping quantization methods, such as Sigma-Delta (Σ∆) modulation and alternative decoding methods (see, e.g., [21,31,6,32,42,17,26,69,27,20,9,36]), have been proposed in the settings of quantization of bandlimited functions, finite frame expansions, and compressed sensing measurements. However, these methods have not been studied in the framework of binary embedding problems.…”

Section: Memoryless Scalar Quantizationmentioning

confidence: 99%

See 1 more Smart Citation

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Huynh

Saab

2019

Comm Pure Appl Math

Self Cite

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

Section: Contributions Related To the Quantization Of Compressed Sensmentioning

confidence: 99%

Section: Memoryless Scalar Quantizationmentioning

confidence: 99%

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Huynh

Saab

2019

Comm Pure Appl Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, if the frame satisfies certain smoothness conditions, the decay rate can be super-linear for first order Σ∆ quantization. Noise shaping schemes for finite frames have also been investigated, some of which yield exponential error decay rate [7,6,8]. In this section, we shall provide necessary information on quantization for finite frames before stating our results in Section 3.…”

Section: Preliminaries On Finite Frame Quantizationmentioning

confidence: 99%

Analysis of Decimation on Finite Frames with Sigma-Delta Quantization

Lin

2019

Constr Approx

View full text Add to dashboard Cite

In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the Σ∆ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. In this study, similar results are proven for finite unitarily generated frames. We introduce a process called alternative decimation on finite frames that is compatible with first and second order Σ∆ quantization. In both cases, alternative decimation results in exponential error decay with respect to the bit usage.

show abstract

“…A number of works also study Σ∆ modulation in combination with compressed sensing [15], [9], [20]. For an overview of Σ∆ modulation in various settings and more general classes of noise shaping methods, we refer the reader to [4].…”

Section: B Sigma-delta Modulation In Mathematical Literaturementioning

confidence: 99%

Higher order 1-bit Sigma-Delta modulation on a circle

Graf

Krahmer

Krause-Solberg

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

Self Cite

View full text Add to dashboard Cite

Manifold models in data analysis and signal processing have become more prominent in recent years. In this paper, we will look at one of the main tasks of modern signal processing, namely, at analog-to-digital (A/D) conversion in connection with a simple manifold model -the circle. We will focus on Sigma-Delta modulation which is a popular method for A/D conversion of bandlimited signals that employs coarse quantization coupled with oversampling. Classical Sigma-Delta schemes provide mismatches and large errors at the initialization point if the signal to be converted is defined on the circle. In this paper, our goal is to get around these problems for Sigma-Delta schemes. Our results show how to design an update for the first and the second order schemes based on the reconstruction error analysis such that for the updated scheme the reconstruction error is improved.

show abstract

Noise-Shaping Quantization Methods for Frame-Based and Compressive Sampling Systems

Cited by 29 publications

References 41 publications

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Analysis of Decimation on Finite Frames with Sigma-Delta Quantization

Higher order 1-bit Sigma-Delta modulation on a circle

Contact Info

Product

Resources

About