Optimal quantizer structure for binary discrete input continuous output channels under an arbitrary quantized-output constraint

TL;DR

This paper characterizes the optimal quantizer structure for binary input channels with continuous output, maximizing mutual information under output constraints, and proposes an efficient algorithm to find the global optimum.

Contribution

It introduces a convex cell structure for optimal quantizers and develops a polynomial-time algorithm for their global optimization.

Findings

Optimal quantizers have convex cell structures in a transformed variable.

A polynomial-time algorithm efficiently finds the global optimal quantizer.

The approach maximizes mutual information under output probability constraints.

Abstract

Given a channel having binary input X = (x_1, x_2) having the probability distribution p_X = (p_{x_1}, p_{x_2}) that is corrupted by a continuous noise to produce a continuous output y \in Y = R. For a given conditional distribution p(y|x_1) = \phi_1(y) and p(y|x_2) = \phi_2(y), one wants to quantize the continuous output y back to the final discrete output Z = (z_1, z_2, ..., z_N) with N \leq 2 such that the mutual information between input and quantized-output I(X; Z) is maximized while the probability of the quantized-output p_Z = (p_{z_1}, p_{z_2}, ..., p_{z_N}) has to satisfy a certain constraint. Consider a new variable r_y=p_{x_1}\phi_1(y)/ (p_{x_1}\phi_1(y)+p_{x_2}\phi_2(y)), we show that the optimal quantizer has a structure of convex cells in the new variable r_y. Based on the convex cells property of the optimal quantizers, a fast algorithm is proposed to find the global optimal quantizer in a polynomial time complexity.