Distributed Scalar Quantization for Computing: High-Resolution Analysis and Extensions

Abstract

Abstract-Communication of quantized information is frequently followed by a computation. We consider situations of distributed functional scalar quantization: distributed scalar quantization of (possibly correlated) sources followed by centralized computation of a function. Under smoothness conditions on the sources and function, companding scalar quantizer designs are developed to minimize mean-squared error (MSE) of the computed function as the quantizer resolution is allowed to grow. Striking improvements over quantizers designed without consideration of the function are possible and are larger in the entropy-constrained setting than in the fixed-rate setting. As extensions to the basic analysis, we characterize a large class of functions for which regular quantization suffices, consider certain functions for which asymptotic optimality is achieved without arbitrarily fine quantization, and allow limited collaboration between source encoders. In the entropy-constrained setting, a single bit per sample communicated between encoders can have an arbitrarily large effect on functional distortion. In contrast, such communication has very little effect in the fixed-rate setting.Index Terms-Asymptotic quantization theory, distributed source coding, optimal point density function, rate-distortion theory.

Communication cost of transforming a nearest plane partition to the Voronoi partition

“…The problem of fine quantization for detection problems is addressed in [15], [2] and [6]. More recently, the design of fine scalar quantizers for distributed function computation with a squared error distortion measure is considered in [12] and succeeding works. Significant benefits, especially in the interactive setting are obtained.…”

Chatting in distributed quantization networks

“…1 without the chatting channel; a comprehensive review of these works and their connections to DFSQ appears in [4]. Similarly, connections to coding for computing (e.g.…”

“…Instead, we consider the complementary asymptotic of high resolution, where the blocklength is considered fixed and the compression rate R is made large [17], [18]. We now introduce high-resolution theory and summarize key results from DFSQ [4], [5].…”

“…Instead, collaboration among encoders, called chatting in this work, can greatly decrease the aggregate communication from the encoders to the decoder to meet a distortion criterion. We explore chatting by building upon the distributed functional scalar quantization (DFSQ) framework, which explores compression through scalar quantization when the decoder desires fidelity in computing a function of the sources [4], [5]. Specifically, we consider the architecture shown in Fig.…”

Distributed Functional Scalar Quantization Simplified

“…The principal previous work in applying high-resolution quantization theory to the acquisition and computation network of Fig. 1 is the distributed functional scalar quantization (DFSQ) framework [2]. The key message from this previous work is that the design of optimal encoders for systems that perform nonlinear computations can be drastically different from what traditional quantization theory suggests.…”