mathematics Article Iterative Algorithm for Parameterization of Two-Region Piecewise Uniform Quantizer for the Laplacian Source Jelena Nikoli´ c 1, * , Danijela Aleksi´ c 2 , Zoran Peri´ c 1 and Milan Dinˇ ci´ c 1   Citation: Nikoli´ c, J.; Aleksi´ c, D.; Peri´ c, Z.; Dinˇ ci´ c, M. Iterative Algorithm for Parameterization of Two-Region Piecewise Uniform Quantizer for the Laplacian Source. Mathematics 2021, 9, 3091. https:// doi.org/10.3390/math9233091 Academic Editor: Galina Bogdanova Received: 11 November 2021 Accepted: 27 November 2021 Published: 30 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Faculty of Electronic Engineering, University of Nis, Aleksandra Medvedeva 14, 18000 Nis, Serbia; zoran.peric@elfak.ni.ac.rs (Z.P.); milan.dincic@elfak.ni.ac.rs (M.D.) 2 Department of Mobile Network Nis, Telekom Srbija, Vozdova 11, 18000 Nis, Serbia; danijelaal@telekom.rs * Correspondence: jelena.nikolic@elfak.ni.ac.rs Abstract: Motivated by the fact that uniform quantization is not suitable for signals having non- uniform probability density functions (pdfs), as the Laplacian pdf is, in this paper we have divided the support region of the quantizer into two disjunctive regions and utilized the simplest uniform quantization with equal bit-rates within both regions. In particular, we assumed a narrow central granular region (CGR) covering the peak of the Laplacian pdf and a wider peripheral granular region (PGR) where the pdf is predominantly tailed. We performed optimization of the widths of CGR and PGR via distortion optimization per border–clipping threshold scaling ratio which resulted in an iterative formula enabling the parametrization of our piecewise uniform quantizer (PWUQ). For medium and high bit-rates, we demonstrated the convenience of our PWUQ over the uniform quantizer, paying special attention to the case where 99.99% of the signal amplitudes belong to the support region or clipping region. We believe that the resulting formulas for PWUQ design and performance assessment are greatly beneficial in neural networks where weights and activations are typically modelled by the Laplacian distribution, and where uniform quantization is commonly used to decrease memory footprint. Keywords: uniform quantization; piecewise uniform quantizer; border threshold; clipping threshold 1. Introduction One of the growing interests in neural networks (NNs) is directed towards the efficient representation of weights and activations by means of quantization [1–14]. Quantization, as a bit-width compression method, is a desirable mechanism that can dictate the entire NN performance [10,12]. In other words, the overall network complexity reduction, provided by the quantization process, can lead to commensurately reduced overall accuracy if the pathway toward this reduction is not chosen prudently. Quantization is significantly bene- ficial for NN implementation on resource-limited devices since it is capable of fitting the whole NN model into the on-chip memory of edge devices such that the high overhead that occurs by off-chip memory access can be mitigated [9]. Namely, standard implementation of NNs supposes 32-bits full-precision (FP32) representation of NN parameters, requir- ing complex and expensive hardware. By quantizing FP32 weights and activations with low-bits, that is, by thoughtfully choosing a quantizer model for NN parameters, one can significantly reduce the required bit-width for the digital representation of NN parameters, greatly reducing the overall complexity of the NN while degrading the network accuracy to some extent [2,3,5,6,8,9]. For that reason, a few of new quantizer models and quantization methodologies have been proposed, for instance in [4,5,11,13], with a main objective—to enable quantized NNs to have the slightly degraded or almost the same accuracy level as their full-precision counterparts. In general, to optimize a quantizer model, one has to know the statistical distribution of the input signal, allowing for the quantizer to be adapted as best as possible to the statistical characteristics of the signal itself. The symmetric Laplacian probability density Mathematics 2021, 9, 3091. https://doi.org/10.3390/math9233091 https://www.mdpi.com/journal/mathematics