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Memoryless scalar quantization for random frames

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Abstract

Memoryless scalar quantization (MSQ) is a common technique to quantize generalized linear samples of signals. The non-linear nature of quantization makes the analysis of the corresponding approximation error challenging, often resulting in the use of a simplifying assumption, called the “white noise hypothesis” (WNH) that is useful, yet also known to be not rigorous and, at least in certain cases, not valid. We obtain rigorous reconstruction error estimates without relying on the WNH in the setting where generalized samples of a fixed deterministic signal are obtained using (the analysis matrix of) a random frame with independent isotropic sub-Gaussian rows; quantized using MSQ; and reconstructed linearly. We establish non-asymptotic error bounds that explain the observed error decay rate as the number of measurements grows, which in the special case of Gaussian random frames show that the error approaches a (small) non-zero constant lower bound. We also extend our methodology to dithered and noisy settings as well as the compressed sensing setting where we obtain rigorous error bounds that agree with empirical observations, again, without resorting to the WNH.

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Notes

  1. It is worth to note, though, that when \(y-Q(y)\) is “shaped”, other left inverses such as “Sobolev duals” may yield significantly better reconstructions, e.g., [5, 19, 21, 33, 38].

  2. Note that one-bit MSQ provides a yet simpler quantization method. The analysis of the one-bit MSQ turns out to be significantly different from the multi-bit MSQ and is not within the scope of this paper, e.g., see [1, 8, 43].

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Acknowledgements

This work was funded in part by a UBC 4YF Fellowship (KM), an NSERC Discovery Accelerator Award (OY), and an NSERC Discovery Grant (OY).

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  1. The University of British Columbia, Vancouver, BC, Canada

    Kateryna Melnykova & Özgür Yilmaz

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  1. Kateryna Melnykova
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Correspondence to Kateryna Melnykova.

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Communicated by Alex Powell.

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Melnykova, K., Yilmaz, Ö. Memoryless scalar quantization for random frames. Sampl. Theory Signal Process. Data Anal. 19, 12 (2021). https://doi.org/10.1007/s43670-021-00012-4

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