# Information-Distilling Quantizers

## TL;DR

This paper investigates the design of scalar quantizers that maximize mutual information between dependent variables, providing bounds and properties especially in low mutual information regimes, with implications for information preservation.

## Contribution

It introduces fundamental bounds and properties for information-distilling quantizers, particularly in low mutual information regimes, connecting to log-loss distortion.

## Key findings

- For binary X, a constant fraction of mutual information can be preserved with O(log(1/I(X;Y))) levels.
- Existence of distributions requiring this many levels to preserve mutual information.
- For larger alphabets, a fraction of mutual information can be preserved with polynomially many levels.

## 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.