Information-Distilling Quantizers

Abstract

Let $X$ and $Y$ be dependent random variables. This paper considers the problem of designing a scalar quantizer for $Y$ to maximize the mutual information between the quantizer's output and $X$, and develops fundamental properties and bounds for this form of quantization, which is connected to the log-loss distortion criterion. The main focus is the regime of low $I(X;Y)$, where it is shown that, if $X$ is binary, a constant fraction of the mutual information can always be preserved using $\mathcal{O}(\log (1/I(X;Y)))$ quantization levels, and there exist distributions for which this many quantization levels are necessary. Furthermore, for larger finite alphabets $2<|\mathcal{X}|<\infty$, it is established that an $\eta$-fraction of the mutual information can be preserved using roughly $(\log (|\mathcal{X}|/I(X;Y)){)}^{\eta \cdot (|\mathcal{X}|-1)}$ quantization levels.