Nonlinear Non-Gaussian Density Steering with Input and Noise Channel Mismatch: Sinkhorn with Memory for Solving the Control-affine Schrödinger Bridge Problem

2026-04-25

AI Summary

This work addresses the control-affine Schrödinger bridge problem under mismatched control and noise channels, a setting where the classical Sinkhorn algorithm fails due to the nonlinearity introduced by the Hopf–Cole transformation in the resulting PDE. To overcome this limitation, the authors propose a novel Sinkhorn-type iterative algorithm equipped with a memory mechanism. By integrating control-affine diffusion modeling, structural analysis of nonlinear boundary-coupled PDEs, and an enhanced dynamic recursion scheme, the method effectively solves the optimal steering problem for non-Gaussian densities. This approach represents the first breakthrough that removes the restrictive channel-matching assumption, establishes local stability guarantees, and demonstrates strong convergence and efficacy in numerical experiments.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Solutions to the Schrödinger bridge problem and its generalizations yield feedback control policies for optimal density steering over a controlled diffusion. To numerically compute the same, the dynamic Sinkhorn recursion has become a standard approach. The mathematical engine behind this approach is the Hopf-Cole transform that recasts the conditions for optimality into a system of boundary-coupled linear PDEs. Recent works pointed out that for the control-affine Schrödinger bridge problem, this exact linearity via Hopf-Cole transform, and thus the standard Sinkhorn recursion, apply only if the control and noise channels are proportional. When the channels do not match, the Hopf-Cole-transformed PDEs remain nonlinear, and no algorithm is available to solve the same. We advance the state-of-the-art by designing a Sinkhorn recursion with memory that leverages the structure of these nonlinear PDEs, and demonstrate how it solves the control-affine Schrödinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.

Problem

• Schrödinger bridge
• density steering
• channel mismatch
• nonlinear PDEs
• control-affine systems

Innovation

• Sinkhorn with memory
• Schrödinger bridge problem
• control-affine systems
• channel mismatch
• nonlinear PDEs