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

TL;DR

This paper develops high-resolution scalar quantizer designs for distributed sources to optimize the accuracy of computed functions, demonstrating significant improvements over traditional methods, especially in entropy-constrained scenarios.

Contribution

It introduces companding scalar quantizer designs tailored for distributed functional computation, extending analysis to various functions and encoder collaboration scenarios.

Findings

Quantizer designs significantly reduce mean-squared error in function computation.

Limited encoder communication can greatly improve performance in entropy-constrained settings.

Regular quantization suffices for a large class of functions, with some achieving asymptotic optimality without fine quantization.

Abstract

Communication of quantized information is frequently followed by a computation. We consider situations of \emph{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.