# Mutual Information Optimal Density Control of Linear Systems and Generalized Schr"odinger Bridges with Reference Refinement

arXiv: https://arxiv.org/abs/2605.09349 · HTML: https://mlsanity.com/paper/2605.09349/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 37 references that are in the map.
- First posted on arXiv 10 May 2026.

## Builds on (most cited first, 23 of 23)

- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [Provably Efficient Maximum Entropy Exploration](https://mlsanity.com/paper/1812.02690/) (2018)
- [Maximum Entropy RL (Provably) Solves Some Robust RL Problems](https://mlsanity.com/paper/2103.06257/) (2021)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Optimal steering of a linear stochastic system to a final probability distribution](https://mlsanity.com/paper/1408.2222/) (2014)
- [Mutual-Information Regularization in Markov Decision Processes and Actor-Critic Learning](https://mlsanity.com/paper/1909.05950/) (2019)
- [Density control of interacting agent systems](https://mlsanity.com/paper/2108.07342/) (2021)
- [Optimal Covariance Control for Stochastic Systems Under Chance Constraints](https://mlsanity.com/paper/1804.02829/) (2018)
- [Nonlinear Covariance Control via Differential Dynamic Programming](https://mlsanity.com/paper/1911.09283/) (2019)
- [Deep RL With Information Constrained Policies: Generalization in Continuous Control](https://mlsanity.com/paper/2010.04646/) (2020)
- [Privacy-Constrained Policies via Mutual Information Regularized Policy Gradients](https://mlsanity.com/paper/2012.15019/) (2020)
- [Maximum entropy optimal density control of discrete-time linear systems and Schr"odinger bridges](https://mlsanity.com/paper/2204.05263/) (2022)
- [Maximum Entropy Density Control of Discrete-Time Linear Systems with Quadratic Cost](https://mlsanity.com/paper/2309.10662/) (2023)
- [Optimal Transport over Deterministic Discrete-time Nonlinear Systems using Stochastic Feedback Laws](https://mlsanity.com/paper/1804.03108/) (2018)
- [Reachability and Controllability Analysis of the State Covariance for Linear Stochastic Systems](https://mlsanity.com/paper/2406.14740/) (2024)
- [Optimal Covariance Steering for Discrete-Time Linear Stochastic Systems](https://mlsanity.com/paper/2211.00618/) (2022)
- [Mutual Information Optimal Control of Discrete-Time Linear Systems](https://mlsanity.com/paper/2507.04712/) (2025)
- [On Policy Stochasticity in Mutual Information Optimal Control of Linear Systems](https://mlsanity.com/paper/2507.21543/) (2025)

## Abstract (arXiv)

We consider a mutual information (MI) regularized optimal density control of a discrete-time linear system. MI optimal control has been utilized for exploration in reinforcement learning and privacy protection in control. MI regularization induces stochasticity in the policy, which poses challenges for applications of MI optimal control in safety-critical scenarios. To remedy this situation, we impose Gaussian density constraints at specified times to directly control state uncertainty. For this MI optimal density control problem, we propose an alternating optimization algorithm and investigate its convergence properties. In addition, we reveal a relationship between the MI optimal density control problem and a so-called generalized Schr"odinger bridge problem associated with the discrete-time linear system. Based on the results of MI optimal density control and this relationship, we also investigate alternating optimization for the Schr"odinger bridge problem and its convergence properties.