Mass-covering behavior induced by forward-KL Markovian projection

Prove that the mass-covering behavior observed in the learned sampler arises directly from optimizing the forward Kullback–Leibler Markovian projection u^* = argmin_{u ∈ U} KL(Π^* | P^u), thereby establishing that the fixed-point iteration inherently yields mass-covering rather than mode-seeking solutions.

Background

In peptide system experiments, the authors report mode-covering behavior, contrasting with mode-seeking tendencies often associated with reverse KL objectives. They link this property to their fixed-point characterization, which corresponds to a forward KL (Π^* | P^u) objective via the Markovian projection.

They explicitly conjecture a causal connection: that the mass-covering property is a direct consequence of the fixed-point iteration’s forward-KL formulation. Formalizing this link would clarify the behavior of matching-based samplers and guide objective selection.