HGR maximal correlation and the ratio of mutual informations

From one of the presentation of Zhao and Chia at Allerton this year, I was made aware of a paper by Elza Erkip and Tom Cover on “The efficiency of investment information” that uses one of my favorite quantities, the Hirschfeld–Gebelein–Rényi maximal correlation; I first discovered it in this gem of a paper by Witsenhausen.

The Hirschfeld–Gebelein–Rényi maximal correlation \rho_m(X,Y) between two random variables X and Y is

\sup_{f \in \mathcal{F}_X, g \in \mathcal{G}_Y} \mathbb{E}[ f(X) g(Y) ]

where \mathcal{F}_X is all real-valued functions such that \mathbb{E}[ f(X) ] = 0 and \mathbb{E}[ f(X)^2 ] = 1 and \mathcal{G}_Y is all real valued functions such that \mathbb{E}[ g(Y) ] = 0 and \mathbb{E}[ g(Y)^2 ] = 1. It’s a cool measure of dependence that covers discrete and continuous variables, since they all get passed through these “normalizing” f and g functions.

The fact in the Erkip-Cover paper is this one:

sup_{ P(z|y) : Z \to Y \to X } \frac{I(Z ; X)}{I(Z ; Y)} = \rho_m(X,Y)^2.

That is, the square of the HGR maximal correlation is the best (or worst, depending on your perspective) ratio of the two sides in the Data Processing Inequality:

I(Z ; Y) \ge I(Z ; X).

It’s a bit surprising to me that this fact is not as well known. Perhaps it’s because the “data processing” is happening at the front end here (by choosing P(z|y)) and not the actual data processing Y \to X which is given to you.

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