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On the Equivalence of Stochastic Control and Path Space Formulations for Schr\"odinger Bridges over Compact Connected Lie Groups

Hamza Mahmood, Georgiy A. Bondar, Abhishek Halder, Adeel Akhtar

arXiv:2609.13758v1math.OCcs.AIcs.SYeess.SYmath.PRstat.ML

TL;DR

The path-space formulation and large-deviation interpretation of Schrödinger bridges on compact connected Lie groups were not addressed by the earlier Lie-group formulation. Using horizontal lift and stochastic anti-development, the paper establishes equivalence between stochastic control and path-space relative entropy minimization, with consequences for existence-uniqueness, interpretation, and computation.

  • Problem

    The earlier Lie-group SBP formulation did not address its path-space formulation or the associated large-deviation interpretation.

    Source: §I · p. 1

  • Method

    Horizontal lift and stochastic anti-development support a Girsanov-type change of measure and relate controlled diffusion on the Lie group to Euclidean coordinates.

    Source: §I · p. 1, D. Girsanov Type Change of Measure · p. 3

  • Results

    Expected control energy equals the relative entropy of the controlled path law with respect to reference Wiener measure, yielding an equivalent path-space SBP.

    Source: §I · p. 1, §V · p. 6

  • Takeaways & Limitations

    The equivalence supports existence-uniqueness, a most-probable-deviation interpretation, and static Sinkhorn computation of the optimal path measure.

    Source: §I · p. 1, A. Large-Deviation Interpretation · p. 5, §III · p. 5

Abstract

from arXiv · show

We establish the equivalence between the stochastic optimal control and path space formulations of the Schr\"odinger 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 $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP

I. INTRODUCTION

The paper addresses the missing path-space formulation and large-deviation interpretation of Schrödinger bridges on compact connected Lie groups. It uses geometric tools to establish equivalence between stochastic control and path-space relative-entropy formulations.

I. INTRODUCTION

  • The earlier Lie-group formulation established existence-uniqueness and dynamic Sinkhorn computation but left the path-space formulation and large-deviation interpretation unaddressed.The latter interprets the bridge as the most probable deviation of uncontrolled dynamics consistent with endpoint constraints.Source: §I · p. 1
  • Horizontal lift and stochastic anti-development represent the diffusion in Euclidean coordinates, enabling an intrinsic change of measure despite varying tangent spaces.The paper contrasts this setting with Lie-group approaches based on logarithm and exponential maps.Source: §I · p. 1
  • The paper derives a Girsanov-type change of measure and equates expected control energy with relative entropy of the controlled path law against reference Wiener measure.This recasts the stochastic-control SBP as path-space relative-entropy minimization subject to endpoint constraints.Source: §I · p. 1
  • The equivalence supports existence-uniqueness, a most-probable-deviation interpretation, and computation of the optimal path measure using static Sinkhorn recursions.The equivalence is numerically illustrated on the torus T2.Source: §I · p. 1

II. EQUIVALENCE OF THE STOCHASTIC CONTROL AND PATH SPACE FORMULATIONS

  • The paper develops the path-space change-of-measure construction on the Lie group and establishes its equivalence with the control formulation.Source: §II · p. 2

A. Background

The setup considers controlled kinematics on a compact connected Lie group, with controls expressed in Euclidean coordinates and states evolving on the group.

A. Background

  • The state R_t lies in a compact connected Lie group G, whose Lie algebra g is finite-dimensional and identified with R^n through the hat map.Source: A. Background · p. 2
  • Left multiplication transports Lie-algebra elements to the state’s tangent space, and the control input Ω maps group states and time to R^n.Source: A. Background · p. 2
  • The controlled kinematic equation specifies the state’s time derivative in its tangent space using the control input.Source: A. Background · p. 2
  • The stochastic kinematic equation is a controlled Stratonovich SDE driven by standard R^n-valued Brownian motion with isotropic diffusion strength σ>0.Source: A. Background · p. 2

B. The Schr¨odinger Bridge Problem

The Schrödinger bridge problem seeks a finite-energy Markovian control that steers a stochastic state on G between prescribed endpoint densities while minimizing expected control effort.

B. The Schr¨odinger Bridge Problem

C. Path Space Setup and Stochastic Anti-Development

The path-space setup uses a horizontal frame and stochastic anti-development to handle Lie-group diffusion through Euclidean coordinates, under assumptions that support the change-of-measure construction.

C. Path Space Setup and Stochastic Anti-Development

D. Girsanov Type Change of Measure

The paper constructs a Girsanov change of measure for the Lie-group diffusion by using stochastic anti-development to address noise in varying tangent spaces. Under the stated assumptions, the transformed measure gives the controlled diffusion’s path law.

D. Girsanov Type Change of Measure

  • Stochastic anti-development represents the diffusion in Euclidean coordinates, allowing a classical Girsanov change of measure despite noise in varying tangent spaces.The construction also relates the horizontal frame to the left-invariant frame through an orthogonal map.Source: D. Girsanov Type Change of Measure · p. 3
  • Under Assumptions 1–3, the stochastic exponential defines PΩ, under which the transformed noise is standard Rn-valued Brownian motion and R has the controlled-diffusion path law.The Novikov condition ensures the stochastic exponential is a martingale.Source: D. Girsanov Type Change of Measure · p. 3
  • The horizontal and left-invariant representations have the same infinitesimal generator, so well-posedness in law makes their path laws coincide.Consequently, PΩ is the path law of the controlled diffusion.Source: D. Girsanov Type Change of Measure · p. 4
  • The change of measure preserves the initial distribution ρ0.The martingale property conditional on R0 gives the stated preservation.Source: D. Girsanov Type Change of Measure · p. 4

E. Relative Entropy and Control Energy

Under the hypotheses of the Girsanov result and an additional finite-energy condition, Corollary 1 relates control energy to relative entropy of the controlled path law.

E. Relative Entropy and Control Energy

  • Lemma 1 supplies the change-of-measure link used to connect relative entropy with control energy.The section identifies this link as useful for the subsequent development.Source: E. Relative Entropy and Control Energy · p. 4
  • Corollary 1 relates the controlled path law’s KL divergence to expected control energy under the additional finite-energy assumption.The corollary is stated under the hypotheses of Lemma 1.Source: E. Relative Entropy and Control Energy · p. 4
  • Under PΩ, the stochastic-integral term is a square-integrable martingale, supporting the entropy–energy identity.This conclusion uses the additional finite-energy assumption.Source: E. Relative Entropy and Control Energy · p. 4

F. Equivalence of the Two Formulations

The control formulation can be recast as relative entropy minimization over path measures with prescribed endpoint marginals. The controlled diffusion’s path law supplies the measure associated with each admissible control.

F. Equivalence of the Two Formulations

  • The reference Wiener measure is defined on continuous paths in G over [0, 1].The path space is X = C([0, 1]; G).Source: F. Equivalence of the Two Formulations · p. 4
  • For an admissible pair, PΩ denotes the controlled diffusion’s path law, and its time-t marginals agree with the state densities.The reference measure is the Wiener measure induced by the uncontrolled diffusion.Source: F. Equivalence of the Two Formulations · p. 4
  • Because admissible measures have the prescribed endpoint marginals, minimizing the control objective is equivalent to minimizing relative entropy over those path measures.The equivalence uses the control-energy and relative-entropy identity.Source: F. Equivalence of the Two Formulations · pp. 4–5

III. EXISTENCE-UNIQUENESS OF THE SBP SOLUTION

For compact connected Lie groups and the stated endpoint-density and control assumptions, the SBP has a unique minimizing pair. The path-space proof establishes existence and uniqueness directly, and its minimizer corresponds to an admissible controlled diffusion.

III. EXISTENCE-UNIQUENESS OF THE SBP SOLUTION

  • The SBP admits a unique minimizing pair under the theorem’s endpoint-density and admissible-control assumptions.The endpoint densities are strictly positive and belong to P2(G) ∩ C2(G); controls satisfy Assumptions 1–3.Source: §III · p. 5
  • The path-space problem admits a minimizer because relative entropy is lower semicontinuous with weakly compact sublevel sets and the endpoint-constrained set is weakly closed.The proof first establishes that the relative entropy problem has finite value.Source: §III · p. 5
  • Strict convexity gives a unique path-space minimizer, which is a Doob h-transform with bounded drift and induces an admissible finite-energy control.The paper concludes that the path-space relaxation is exact.Source: §III · p. 5
  • Unlike the earlier approach in [10, Theorem 1], the existence-uniqueness proof here is direct in the path-space formulation.The earlier proof proceeded through necessary optimality conditions and a Schrödinger system.Source: §III · p. 5

A. Large-Deviation Interpretation

The path-space formulation gives a large-deviation interpretation: among path measures satisfying the endpoint constraints, Popt is the most probable deviation from the uncontrolled dynamics and corresponds to the most likely bridge evolution.

A. Large-Deviation Interpretation

IV. STATIC SINKHORN AND NUMERICAL EXAMPLE

The torus example compares the bridge obtained from the stochastic-control formulation with the path-space minimizer computed using static Sinkhorn recursions.

IV. STATIC SINKHORN AND NUMERICAL EXAMPLE

  • The numerical example uses antipodal unimodal reference measures supported on the torus T2.Source: §IV · p. 6
  • The 50 sampled data points used for (32) track exactly the continuous density function that solves (7).The paper reports that the two formulations yield the same Schrödinger bridge between the reference measures.Source: §IV · p. 6
  • For (7), the authors discretize T2 on a 128 × 128 grid and solve the boundary-coupled heat PDE system using dynamic Sinkhorn recursions.Source: §IV · p. 6
  • For (32), strict convexity and strong duality enable static Sinkhorn recursions over constraint multipliers; the implementation samples N = 50 points from each endpoint measure.The recursions are reported to converge linearly to a unique set of multipliers, from which Popt is obtained.Source: §IV · p. 6

V. CONCLUDING REMARKS

The paper establishes equivalence between the stochastic optimal-control and path-space formulations of the Schrödinger bridge problem on a compact connected Lie group. The equivalence relates expected control energy to path-space relative entropy and has analytic, probabilistic, and computational implications.

V. CONCLUDING REMARKS

  • The equivalence relates expected control energy to path-space relative entropy and supports existence and uniqueness of the bridge, a large-deviation interpretation, and static Sinkhorn computation of the optimal path measure.Source: §V · p. 6
  • Horizontal lift and stochastic anti-development provide the geometric tools for deriving the Girsanov-type change of measure on the Lie group.Source: §V · p. 6
  • Scattered endpoint samples are a numerical convenience for static Sinkhorn; the same fixed grid used for (7) can also be used for (32).Source: §V · p. 6
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