Distributed Noise-Shaping Quantization: II. Classical Frames

Chou, Evan; Güntürk, C. Sınan

doi:10.1007/978-3-319-54711-4_8

Cited by 13 publications

(15 citation statements)

References 10 publications

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“…The proof of the above is simply by induction and can be found, for example, in [19]. Our algorithm are inspired by the approach to reconstruction from noise-shaping quantized measurements used in [17] and [62] (see also [20]).…”

Section: Noise-shaping Quantization Methodsmentioning

confidence: 99%

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Huynh

Saab

2019

Comm Pure Appl Math

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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.

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Section: Noise-shaping Quantization Methodsmentioning

confidence: 99%

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Huynh

Saab

2019

Comm Pure Appl Math

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“…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%

“…Beta Dual of Distributed Noise Shaping. Chou and Günturk [8,6] proposed a distributed noise shaping quantization scheme with beta duals as an example. The definition of a beta dual is as follows:…”

Section: Finitementioning

confidence: 99%

Analysis of Decimation on Finite Frames with Sigma-Delta Quantization

Lin

2019

Constr Approx

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“…The analysis operator Φ of T u has {φ * j } j as its rows. As symmetry occurs naturally in many applications, it is not surprising that unitarily generated frames receive serious attention, and their applications in signal processing abound, [17,15,7,9].…”

Section: 2mentioning

confidence: 99%

“…Beta Dual of Distributed Noise Shaping. Chou and Günturk [9,7] proposed a distributed noise shaping quantization scheme with beta duals as an example. The definition of a beta dual is as follows:…”

Section: 2mentioning

confidence: 99%

Adapted Decimation on Finite Frames for Arbitrary Orders of Sigma-Delta Quantization

Lin

2019

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

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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. We build on our previous result, which extended signal decimation to finite frames, albeit only up to the second order. In this study, we introduce a new scheme called adapted decimation, which yields polynomial reconstruction error decay rate of arbitrary order with respect to the oversampling ratio, and exponential with respect to the bit-rate. ∞ −∞ f (t)e −2πıtγ dt.The Fourier transform can also be uniquely extended to L 2 (R) as a unitary transformation. An important component of A/D conversion is the following theorem:Theorem 1.2 (Classical Sampling Theorem). Given f ∈ P W [−1/2,1/2] , for any g ∈ L 2 (R) satisfying•ĝ(ω) = 1 on [−1/2, 1/2] •ĝ(ω) = 0 for |ω| ≥ 1/2 + , with > 0 and T ∈ (0, 1 − 2 ), t ∈ R, one haswhere the convergence is both uniform on compact sets of R and in L 2 .2010 Mathematics Subject Classification. 42C15, 94A08, 94A34.

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Distributed Noise-Shaping Quantization: II. Classical Frames

Cited by 13 publications

References 10 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

Adapted Decimation on Finite Frames for Arbitrary Orders of Sigma-Delta Quantization

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