On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups
Authors: Hamza Mahmood, Georgiy A. Bondar, Abhishek Halder, Adeel Akhtar
Organizations: Dept. of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Applied Mathematics at the University of California at Santa Cruz, CA, 95064, USA · Dept. of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA
We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus T2. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP
Figures & tables
Fig. 1: Evolution of the Schrödinger bridge probability density between endpoint distributions ρ0 and ρ1 on the torus T2 . In blue is shown the continuous density function ρσopt(⋅,t) – the solution of ( 7 ) computed by solving the boundary-coupled system of heat PDEs on the Lie group as in [ 10 ] . In orange are the scattered data points transported along Popt – the minimizer of ( 32 ) computed by solving the static Sinkhorn recursions.
This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on SO(2) and SO(3). The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .
Hamza Mahmood, Abhishek Halder, Adeel Akhtar
Department of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA
The Schrödinger bridge owes its computational power to a single structural fact: by Girsanov's theorem the controlled problem is a Kullback--Leibler (KL) projection onto a fixed reference measure, solvable by alternating projections. This letter shows that the fact does not survive risk sensitivity. When the expected path cost is replaced by the entropic risk measure and both endpoint marginals are kept as hard constraints, the resulting fixed-point bridge value Jθ (the soft-problem value at the multiplier that enforces the terminal constraint) admits no representation as a constrained KL minimum against any fixed path-space reference with a regular endpoint law (a class strictly larger than the uniformly elliptic diffusion references: no Markov property is required), even allowing an additive normalisation depending on the initial marginal. Moreover, no single reference generates the one-parameter family in the risk parameter. The obstruction is computed in closed form: the Gaussian bridge value violates, by exactly θ/2, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey. In place of the projection, the theory rests on a terminal-multiplier fixed point and an asymmetric factorisation penalising the score energy of the backward factor.
Controlling the evolution of a many-body stochastic system from a disordered reference state to a structured target ensemble, characterized empirically through samples, arises naturally in non-equilibrium statistical mechanics and stochastic control. The natural relaxation of such a system - driven by diffusion - runs from the structured target toward the disordered reference. The natural question is then: what is the minimum-work stochastic process that reverses this relaxation, given a pathwise cost functional combining spatial penalties and control effort? Computing this optimal process requires knowledge of trajectories that already sample the target ensemble - precisely the object one is trying to construct. We resolve this by establishing a time-reversal duality: the value function governing the hard backward dynamics satisfies an equivalent forward-in-time HJB equation, whose solution can be read off directly from the tractable forward relaxation trajectories. Via the Cole-Hopf transformation and its associated Feynman-Kac representation, this forward potential is computed as a path-space free energy averaged over these forward trajectories - the same relaxation paths that are easy to simulate - without any backward simulation or knowledge of the target beyond samples. The resulting framework provides a physically interpretable description of stochastic transport in terms of path-space free energy, risk-sensitive control, and spatial cost geometry. We illustrate the theory with numerical examples that visualize the learned value function and the induced controlled diffusions, demonstrating how spatial cost fields shape transport geometry analogously to Fermat's Principle in inhomogeneous media. Our results establish a unifying connection between stochastic optimal control, Schrödinger bridge theory, and non-equilibrium statistical mechanics.
Haiqian Yang, Vishaal Krishnan, Sumit Sinha +1
Department of Mechanical Engineering, MIT, Cambridge, MA 02139 · SEAS, Harvard University, Cambridge, MA 02138 · SEAS, Physics and OEB, Harvard University, Cambridge, MA 02138