Variational expressions and Uhlmann theorem for measured divergences

Kun Fang, Hamza Fawzi, Omar Fawzi (Feb 12 2025).

Abstract: We present a variational expression for measured $f$-divergences for convex functions $f$. Under a suitable operator convexity condition, this variational expression leads to a convex optimization problem recovering the known expression in the case of Rényi divergences. As an application, we establish an Uhlmann theorem for a wide class of measured $f$-divergences including measured $\alpha$-Rényi divergences for all $\alpha \geq 0$. The well-known Uhlmann theorem for the fidelity corresponds to the special case $\alpha = \frac{1}{2}$.

Arxiv: https://arxiv.org/abs/2502.07745