The LQR-Schr{ö}dinger Bridge

TL;DR

The paper presents a discrete-time Schrödinger bridge with a pathwise LQR cost, enabling closed-form solutions under Gaussian marginals via backward and forward discrete Riccati equations. By modeling the reference with a pairwise LQR loss, the authors derive a Markovian optimal process with an explicit transition kernel, connecting Doob transforms and Kalman duality. The key contributions include the forward–backward Riccati recursions, explicit kernel updates, and a demonstration that potentials shape non-homogeneous Gaussian transports, extending Bures transport to curved geometries. The results yield efficient construction of complex Gaussian processes with path constraints, and the framework opens avenues for applications in stochastic path planning, covariance steering, and entropy-regularized transport on nontrivial geometries.

Abstract

We consider the Schr{ö}dinger bridge problem in discrete time, where the pathwise cost is replaced by a sum of quadratic functions, taking the form of a linear quadratic regulator (LQR) cost. This cost comprises potential terms that act as attractors and kinetic terms that control the diffusion of the process. When the two boundary marginals are Gaussian, we show that the LQR-Schr{ö}dinger bridge problem can be solved in closed form. We follow the dynamic programming principle, interpreting the Kantorovich potentials as cost-to-go functions. Under the LQR-Gaussian assumption, these potentials can be propagated exactly in a backward and forward passes, leading to a system of dual Riccati equations, well known in estimation and control. This system converges rapidly in practice. We then show that the optimal process is Markovian and compute its transition kernel in closed form as well as the Gaussian marginals. Through numerical experiments, we demonstrate that this approach can be used to construct complex, non-homogeneous Gaussian processes with acceleration and loops, given well-chosen attractive potentials. Moreover, this approach allows extending the Bures transport between Gaussian distributions to more complex geometries with negative curvature.

Figures (4)

• Figure 1: As we increase the potential force, shown in green, the Gaussians' Bures geodesics, shown in blue, are deformed and ultimately reduce to a single point where the information is maximal.
• Figure 2: The figure on the left shows a standard Brownian bridge between two Dirac marginals, which is obtained when we consider a warm temperature $\varepsilon=1$. There is no potential in the LQR cost, such that $q=0$ and $r=100$. The figure on the right shows an optimal transport between two Gaussians, obtained with a cold temperature $\varepsilon=0.001$. We have added a constant potential in the middle of the path, such that $q=0.3$ and $r=10$. The covariances of the Gaussian process are shown in blue at $3 \sigma$ here, whereas the sampled trajectories are shown in black.
• Figure 3: Temperature parameter $\varepsilon=0.001$. From left to right: Wave path following with $r=1, q=10, K=15$, where we also use $K$ waypoints (green ellipsoid has been rescaled for better visualization in this case); Zig-zag case with $r=10, q=0.1, K=100$, where we use $2$ waypoints to steer the Gaussian process;Scoubidou case with $r=10, q=0.2, K=100$, where we use $3$ waypoints; Twister obstacle with $r=10, q=0.2, K=200$, where we use one waypoint with a non-isotropic potential matrix, such that the Gaussian covariances of the marginals are twisted near the potential. The blue ellipsoids show the Gaussian marginals and the green one the potential forces. All the covariances are shown at $1 \sigma$ here.
• Figure 4: Same conditions as in Figure \ref{['Figure2']}, but with the temperature parameter $\varepsilon=1$. The process is no longer deterministic in the Wave path following and is much more diffusive in the other experiments.

Theorems & Definitions (4)

• Remark 1: Connexion with least action principle
• Remark 2: Connexion with Doob’s transform & RL
• Remark 3: Duality of Riccati equations
• Remark 4: Connexion with Bures geodesics