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

arXiv: https://arxiv.org/abs/2604.23370 · HTML: https://mlsanity.com/paper/2604.23370/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 33 references that are in the map.
- First posted on arXiv 25 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Optimal steering of a linear stochastic system to a final probability distribution, part II](https://mlsanity.com/paper/1410.3447/) (2014)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems](https://mlsanity.com/paper/1909.12511/) (2019)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly](https://mlsanity.com/paper/2208.09182/) (2022)
- [Linear Hamilton Jacobi Bellman Equations in High Dimensions](https://mlsanity.com/paper/1404.1089/) (2014)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost](https://mlsanity.com/paper/2407.15245/) (2024)
- [Markov Kernels, Distances and Optimal Control: A Parable of Linear Quadratic Non-Gaussian Distribution Steering](https://mlsanity.com/paper/2504.15753/) (2025)
- [Control-affine Schr"odinger Bridge and Generalized Bohm Potential](https://mlsanity.com/paper/2508.08511/) (2025)

## Built on by (most cited first, 1 of 1)

- [Linear Exponential Quadratic Gaussian Covariance Steering](https://mlsanity.com/paper/2609.12463/) (2026)

## Abstract (arXiv)

Solutions to the Schr"odinger 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 Schr"odinger 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 Schr"odinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.