Annotation: dorsal/arxiv

Authors	Zachary Roberston
Categories	cs.IT math.IT
ArXiv ID	2601.08929vv1
URL	https://arxiv.org/abs/2601.08929
License	http://creativecommons.org/licenses/by/4.0/

Authors

Zachary Roberston

Abstract

We study when the variable-indexed matrix of pairwise $f$-mutual informations $M^{(f)}_{ij}=I_f(X_i;X_j)$ is positive semidefinite (PSD). Let $f:(0,\infty)\to\mathbb{R}$ be convex with $f(1)=0$, finite in a neighborhood of $1$, and with $f(0)<\infty$ so that diagonal terms are finite. We give a sharp \emph{local} characterization around independence: there exists $\delta=\delta(f)>0$ such that for every $n$ and every finite-alphabet family $(X_1,\ldots,X_n)$ whose pairwise joint-to-product ratios lie in $(1-\delta,1+\delta)$, the matrix $M^{(f)}$ is PSD if and only if $f$ is analytic at $1$ with a convergent expansion $f(t)=\sum_{m=2}^{\infty} a_m (t-1)^m$ and $a_m\ge 0$ on a neighborhood of $1$. Consequently, any negative Taylor coefficient yields an explicit finite-alphabet counterexample under arbitrarily weak dependence, and non-analytic convex divergences (e.g.\ total variation) are excluded. This PSD requirement is distinct from Hilbertian/metric properties of divergences between distributions (e.g.\ $\sqrt{\mathrm{JS}}$): we study PSD of the \emph{variable-indexed} mutual-information matrix. The proof combines a replica embedding that turns monomial terms into Gram matrices with a replica-forcing reduction to positive-definite dot-product kernels, enabling an application of the Schoenberg--Berg--Christensen--Ressel classification.

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