Convergence and well-posedness of the BMS fixed-point iteration

Establish convergence guarantees for the fixed-point iteration u_{i+1} = Φ(u_i) and its damped variant u_{i+1} = αΦ(u_i) + (1−α)u_i used in the Bridge Matching Sampler to learn the Markovian control field u, and ascertain conditions under which a solution exists and is unique.

Background

The paper introduces the Bridge Matching Sampler (BMS), which learns a stochastic transport map between a prior and an unnormalized target density via a generalized fixed-point iteration alternating reciprocal projections and Markovianization. The iteration u_{i+1} = Φ(u_i) relies on Nelson’s identity and a generalized target score identity, and a damped variant is proposed to improve stability.

While BMS is shown to be effective empirically, the authors explicitly state that theoretical guarantees are lacking: convergence behavior and conditions for existence and uniqueness of solutions to the fixed-point iteration remain unproven. Addressing these questions would provide a rigorous foundation for the method’s stability and reliability.