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 \(δ=δ(f)>0\) such that for every \(n\) and every finite-alphabet family \((X_1,\ldots,X_n)\) whose pairwise joint-to-product ratios lie in \((1-δ,1+δ)\), 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.

翻译：我们研究变量索引的成对$f$-互信息矩阵$M^{(f)}_{ij}=I_f(X_i;X_j)$在何种条件下为半正定矩阵。设$f:(0,\infty)\to\mathbb{R}$为凸函数，满足$f(1)=0$，在$1$的邻域内有限，且$f(0)<\infty$以保证对角项有限。我们给出了独立性邻域内的精确局部刻画：存在$\delta=\delta(f)>0$，使得对于任意$n$及任意有限字母表变量族$(X_1,\ldots,X_n)$，若其成对联合分布与乘积分布的比值位于$(1-\delta,1+\delta)$区间内，则矩阵$M^{(f)}$为半正定矩阵当且仅当$f$在$1$处解析且具有收敛展开式$f(t)=\sum_{m=2}^{\infty} a_m (t-1)^m$，并在$1$的邻域内满足$a_m\ge 0$。由此可得，任何负的泰勒系数都会在任意弱依赖条件下产生显式的有限字母表反例，而非解析的凸散度（如全变差散度）则被排除。该半正定性要求不同于分布间散度的希尔伯特度量性质（例如$\sqrt{\mathrm{JS}}$散度）：我们研究的是变量索引互信息矩阵的半正定性。证明结合了将单项式项转化为格拉姆矩阵的副本嵌入技术，以及通过副本强制约简为正定点积核的方法，从而得以应用Schoenberg--Berg--Christensen--Ressel分类定理。