Distributed Noise-Shaping Quantization: I. Beta Duals of Finite Frames and Near-Optimal Quantization of Random Measurements

Evan Chou, C. Sinan Gunturk

Research output: Contribution to journalArticle

Abstract

This paper introduces a new algorithm for the so-called “analysis problem” in quantization of finite frame representations that provides a near-optimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called distributed noise shaping and, in particular, beta duals of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for random frames, using beta duals results in near-optimally accurate reconstructions with respect to both the frame redundancy and the number of levels at which the frame coefficients are quantized. More specifically, for any frame E of m vectors in (Formula presented.) except possibly from a subset of Gaussian measure exponentially small in m and for any number (Formula presented.) of quantization levels per measurement to be used to encode the unit ball in (Formula presented.), there is an algorithmic quantization scheme and a dual frame together that guarantee a reconstruction error of at most (Formula presented.), where (Formula presented.) can be arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.

Original languageEnglish (US)
Pages (from-to)1-22
Number of pages22
JournalConstructive Approximation
DOIs
StateAccepted/In press - May 23 2016

Fingerprint

Quantization
Redundancy
Dual Frame
Costs
Gaussian Measure
Unit ball
Performance Analysis
Computational Cost
Optimal Solution
Subset
Coefficient

Keywords

  • Beta encoding
  • Finite frames
  • Noise shaping
  • Quantization
  • Random matrices

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Computational Mathematics

Cite this

@article{10ec559c2d014a80992449d5fc54d3a3,
title = "Distributed Noise-Shaping Quantization: I. Beta Duals of Finite Frames and Near-Optimal Quantization of Random Measurements",
abstract = "This paper introduces a new algorithm for the so-called “analysis problem” in quantization of finite frame representations that provides a near-optimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called distributed noise shaping and, in particular, beta duals of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for random frames, using beta duals results in near-optimally accurate reconstructions with respect to both the frame redundancy and the number of levels at which the frame coefficients are quantized. More specifically, for any frame E of m vectors in (Formula presented.) except possibly from a subset of Gaussian measure exponentially small in m and for any number (Formula presented.) of quantization levels per measurement to be used to encode the unit ball in (Formula presented.), there is an algorithmic quantization scheme and a dual frame together that guarantee a reconstruction error of at most (Formula presented.), where (Formula presented.) can be arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.",
keywords = "Beta encoding, Finite frames, Noise shaping, Quantization, Random matrices",
author = "Evan Chou and Gunturk, {C. Sinan}",
year = "2016",
month = "5",
day = "23",
doi = "10.1007/s00365-016-9344-4",
language = "English (US)",
pages = "1--22",
journal = "Constructive Approximation",
issn = "0176-4276",
publisher = "Springer New York",

}

TY - JOUR

T1 - Distributed Noise-Shaping Quantization

T2 - I. Beta Duals of Finite Frames and Near-Optimal Quantization of Random Measurements

AU - Chou, Evan

AU - Gunturk, C. Sinan

PY - 2016/5/23

Y1 - 2016/5/23

N2 - This paper introduces a new algorithm for the so-called “analysis problem” in quantization of finite frame representations that provides a near-optimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called distributed noise shaping and, in particular, beta duals of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for random frames, using beta duals results in near-optimally accurate reconstructions with respect to both the frame redundancy and the number of levels at which the frame coefficients are quantized. More specifically, for any frame E of m vectors in (Formula presented.) except possibly from a subset of Gaussian measure exponentially small in m and for any number (Formula presented.) of quantization levels per measurement to be used to encode the unit ball in (Formula presented.), there is an algorithmic quantization scheme and a dual frame together that guarantee a reconstruction error of at most (Formula presented.), where (Formula presented.) can be arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.

AB - This paper introduces a new algorithm for the so-called “analysis problem” in quantization of finite frame representations that provides a near-optimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called distributed noise shaping and, in particular, beta duals of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for random frames, using beta duals results in near-optimally accurate reconstructions with respect to both the frame redundancy and the number of levels at which the frame coefficients are quantized. More specifically, for any frame E of m vectors in (Formula presented.) except possibly from a subset of Gaussian measure exponentially small in m and for any number (Formula presented.) of quantization levels per measurement to be used to encode the unit ball in (Formula presented.), there is an algorithmic quantization scheme and a dual frame together that guarantee a reconstruction error of at most (Formula presented.), where (Formula presented.) can be arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.

KW - Beta encoding

KW - Finite frames

KW - Noise shaping

KW - Quantization

KW - Random matrices

UR - http://www.scopus.com/inward/record.url?scp=84969821511&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969821511&partnerID=8YFLogxK

U2 - 10.1007/s00365-016-9344-4

DO - 10.1007/s00365-016-9344-4

M3 - Article

AN - SCOPUS:84969821511

SP - 1

EP - 22

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

ER -