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 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 that the frame coefficients are quantized at. More specifically, for any frame E of m vectors in R k except possibly from a subset of Gaussian measure exponentially small in m and for any number L ≥ 2 of quantization levels per measurement to be used to encode the unit ball in R k , there is an algorithmic quantization scheme and a dual frame together which guarantee a reconstruction error of at most √ kL −(1−η)m/k , where η can be arbitrarily small for sufficiently large problems. Additional features of the proposed algorithm include low computational cost and parallel implementability.

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

“…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 [8,7,9]. In this section, we shall provide necessary information on quantization for finite frames before stating our results in Section 3.…”

“…Σ∆ quantization is a subclass of the more general noise shaping quantization, where the quantization scheme is designed such that the reconstruction error is easily separated from the true signal in the frequency domain. For instance, it is pointed out in [9] that the reconstruction error of Σ∆ quantization for bandlimited functions is concentrated in high frequency ranges. Since audio signals have finite bandwidth, it is then possible to separate the signal from the error using low-pass filters.…”

“…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].…”

“…Using smooth frame-path with vanishing derivatives at the endpoint yields polynomial error decay rate for higher order Σ∆ quantization. By carefully matching between the dual frame and the quantization scheme, [9] proved that using β-dual for random frames will result in exponential decay with near-optimal exponent and high probability.…”

“…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:…”