Lifted Schrödinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions

2026-05-23

AI Summary

This work addresses the stochastic density control problem with Brownian motion prior and Gaussian mixture endpoints—i.e., Schrödinger bridges lacking closed-form solutions—by proposing an augmented path-space framework. By introducing source–target component labels, the original problem is decomposed into pairwise Schrödinger bridges between Gaussian components and recast as a finite-dimensional entropic optimal transport problem. The information loss incurred by projecting back onto unlabeled paths is analyzed, revealing a non-negative relative entropy gap between the lifted and original bridges; this gap vanishes under a shared-path-potential condition. Leveraging the lifted bridge, Sinkhorn iterations, and posterior mean drift derivation, the authors obtain explicit expressions for marginal distributions, drift fields, and an upper bound on kinetic energy. Numerical experiments demonstrate effective control over both density evolution and shape.

Abstract

We study stochastic density control between Gaussian-mixture endpoint distributions under Brownian prior dynamics. Since the direct Schrödinger bridge between Gaussian mixtures is generally not available in closed form, we introduce a lifted path-space construction in which each trajectory is augmented with a source--target component label. Consequently, the problem decomposes into Gaussian component-to-component Schrödinger bridges with explicit marginal, drift, and cost formulas, while the mixture-level assignment reduces to a finite-dimensional entropic coupling problem with a Sinkhorn scaling form. We then analyze the projection obtained by discarding or forgetting the label. By construction, the projected law satisfies the original Gaussian-mixture endpoint constraints, but its relative entropy generally differs from the lifted relative entropy by a nonnegative conditional label-information gap. This gap reveals a path-space obstruction: the lifted optimizer cannot, in general, be identified with the direct unlabeled Schrödinger bridge after projection. We also derive the posterior-averaged Markov drift associated with the projected marginal flow, prove a kinetic-energy upper bound, and identify a common path-potential condition under which the projection gap vanishes. Several numerical illustrations showing density and shape control are recorded for a self-contained exposition.

Problem

• Schrödinger bridges
• Gaussian mixtures
• stochastic control
• path-space obstructions
• projection gaps

Innovation

• Lifted Schrödinger Bridge
• Gaussian Mixture
• Entropic Coupling
• Path-Space Obstruction
• Sinkhorn Scaling