Closed-form closest product chain under reverse KL and α-divergences

Derive a closed-form expression for the closest product transition matrix that minimizes D_f^π(P || ⊗_{i=1}^d L_i) when f generates either the reverse Kullback–Leibler divergence (f(t) = −ln t) or the α-divergence (f(t) = (t^α−1)/(α−1)); specifically, obtain the joint optimizer over all component kernels L_i ∈ ℒ(𝓧^{(i)}) rather than only solutions with prescribed marginals.

Background

For KL divergence, the paper provides a closed-form closest product chain as the tensor of marginal transition matrices and proves a Pythagorean identity. For reverse KL and α-divergences, the authors characterize one-step updates with prescribed marginals and propose a coordinate descent algorithm, but do not obtain a global closed-form minimizer.

A closed-form solution would generalize matrix nearness results to broader divergences and strengthen large-deviation and mixing-time applications beyond the KL setting.