Distributed noise-shaping quantization: I. Beta duals of finite frames and near-optimal quantization of random measurements

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Abstract

This paper introduces a new algorithm for the so-called "Analysis Problem" in quantization of finite frame representations which provides a near-optimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called {\em distributed noise-shaping}, and in particular, {\em beta duals} of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for Gaussian random frames, using beta duals result in near-optimally accurate reconstructions with respect to both the frame redundancy and the number of levels that the frame coefficients are quantized at. More specifically, if $L$ quantization levels per measurement are used to encode the unit ball in $\mathbb{R}^k$ via a Gaussian frame of $m$ vectors, then with overwhelming probability the beta-dual reconstruction error is shown to be bounded by $\sqrt{k}L^{-(1-\eta)m/k}$ where $\eta$ is arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.
Original language Undefined 1405.4628 arXiv Published - May 19 2014

• cs.IT
• math.IT

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In: arXiv, 19.05.2014.

Research output: Contribution to journalArticle

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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 which provides a near-optimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called {\em distributed noise-shaping}, and in particular, {\em beta duals} of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for Gaussian random frames, using beta duals result in near-optimally accurate reconstructions with respect to both the frame redundancy and the number of levels that the frame coefficients are quantized at. More specifically, if $L$ quantization levels per measurement are used to encode the unit ball in $\mathbb{R}^k$ via a Gaussian frame of $m$ vectors, then with overwhelming probability the beta-dual reconstruction error is shown to be bounded by $\sqrt{k}L^{-(1-\eta)m/k}$ where $\eta$ is arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.",
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