On One-Bit Quantization

TL;DR

This paper investigates the optimal design of one-bit quantizers for sources in Hilbert spaces, providing methods to identify optimal projections and demonstrating their application to low-dimensional continuous-time processes, with neural network-based training.

Contribution

It introduces a method to determine the optimal projection direction for one-bit quantization and applies it to low-dimensional continuous-time processes using neural networks.

Findings

Optimal quantizer is a projection followed by thresholding.

Neural network-based compressor can learn the optimal quantizer.

Method effectively characterizes low-dimensional structure in continuous-time processes.

Abstract

We consider the one-bit quantizer that minimizes the mean squared error for a source living in a real Hilbert space. The optimal quantizer is a projection followed by a thresholding operation, and we provide methods for identifying the optimal direction along which to project. As an application of our methods, we characterize the optimal one-bit quantizer for a continuous-time random process that exhibits low-dimensional structure. We numerically show that this optimal quantizer is found by a neural-network-based compressor trained via stochastic gradient descent.