Steering probability distributions on curved spaces
Most optimization problems in control theory assume you're working in flat space. You have a system, you want to steer it from one state to another, and you minimize some cost along the way. This works beautifully when your state space is Euclidean, but reality is messier. Rotations don't live in flat space. Neither do orientations, pose estimates, or any number of problems involving symmetry and geometry. When you try to force these curved problems into flat coordinates, you introduce artificial distortions that propagate through your entire solution.
This paper tackles a specific but fundamental version of this problem: how do you steer a probability distribution from one shape to another when you're constrained to live on a curved geometric space, specifically a Lie group? The answer is to stop fighting the geometry and start using it.
The core problem
Imagine you have a probability distribution spread across some space, and another distribution at a different location. You want to morph the first into the second using minimal control effort. On a flat surface, this is already interesting. On a sphere, or the space of 3D rotations, it's more subtle. The paths you can take, the notion of "straight line," the meaning of moving smoothly, all depend on the underlying geometric structure.
Most approaches would handle this by embedding the curved space into higher-dimensional Euclidean space and treating it as a constrained problem there. This is pragmatic but conceptually awkward. It's like trying to solve a navigation problem on Earth using a flat map that preserves area but distorts distances. You can make it work, but you're solving the wrong problem.
The cleaner approach is to formulate the control problem directly on the curved space itself, using only concepts that make sense intrinsically. This is what "coordinate-free" means. You're not choosing a coordinate system (like latitude and longitude) to describe your manifold. Instead, you write equations that respect the manifold's own structure, independent of any labeling system humans might impose.
Why Lie groups matter
Lie groups are special. They're curved spaces with symmetry baked in. Every point has the same local geometry, just oriented differently. This symmetry is powerful for control problems because you can write a single set of equations that works everywhere at once, without needing different formulas for different regions.
The groups studied here, SO(2) and SO(3), are concrete and important. SO(2) is the circle, the space of rotations in 2D. Every point on a circle has identical local structure, which is why you can write down equations that work for all rotation angles simultaneously. SO(3) is the group of 3D rotations, the rotation matrices. It's harder to visualize but follows the same principle. The equations that describe optimal control on SO(3) don't need to be different at different rotations, because the group itself is rotationally invariant.
This property matters immensely for control theory. It means you're not patching together solutions from different coordinate charts. You have one coherent formulation that respects the full symmetry of your problem space.
The steering objective
The goal is to steer a controlled diffusion process from an initial probability distribution to a terminal one, while minimizing control effort. This is called the Schrodinger bridge problem, and it captures a natural optimization principle: among all the ways to morph one distribution into another, find the gentlest one.
Diffusion here means the system has both deterministic control (your steering) and random noise. The randomness is intrinsic, not artificial. In many physical systems, randomness is always present, and pretending it away creates unrealistic models. The Schrodinger bridge framework says, "Given that randomness exists, what's the minimum control effort needed to guide the distribution where you want it to go?"
This is fundamentally different from classical optimal control, which finds the best trajectory for a single particle. Here you're shaping an entire probability cloud, taking into account the diffusive spreading that occurs naturally as the system evolves.
Formulation that respects geometry
The paper develops a coordinate-free formulation of this problem on compact connected Lie groups. This means the mathematical objects used, and the equations they satisfy, are defined intrinsically on the group itself. They don't reference any choice of coordinates or embedding.
What does this look like in practice? Instead of writing down equations in terms of coordinates (x1, x2, ..., xn), you write them in terms of geometric objects that have meaning on the manifold itself. On a Lie group, this includes the group multiplication operation and the Lie algebra, which captures the group's infinitesimal structure. These objects exist whether or not you've assigned coordinates to anything.
The paper establishes existence and uniqueness of solutions to the corresponding Schrodinger system. This is reassuring: the optimal controller you're looking for actually exists, and it's unique. There's no ambiguity about which solution you should use.
Computing the optimal controller
Once you know an optimal controller exists, you need to find it. The paper provides a constructive method using Sinkhorn recursion, an iterative algorithm that refines your approximation of the optimal controller at each step.
The intuition is straightforward. You start with a guess. Then you iteratively improve it by enforcing the optimality conditions more and more strictly. Each iteration brings you closer to a solution that satisfies all the mathematical constraints defining optimality. The algorithm converges to the true optimal controller.
Figure 2 shows this convergence in action, with the inset zooming in on how the iterative refinement proceeds. Each step makes the controller slightly more optimal, and the algorithm terminates when further iterations produce negligible change. This is constructive in the best sense: it doesn't just prove optimality exists, it gives you a concrete procedure to compute it.
Convergence of the iterative refinement algorithm applied to SO(3), showing probability densities at intermediate times (main plot) and the convergence behavior of the Sinkhorn recursion (inset)
SO(2): The circular case
The simplest example is SO(2), the circle. Here the geometry is intuitive. A probability distribution on a circle is just a density that integrates to one around the circle's circumference. Steering one distribution to another means moving probability mass around the circle to change the shape.
Figure 1 illustrates this with two nearby peaks that gradually morph into each other as time progresses from 0 to 1. You can actually see the optimal control at work: the distribution doesn't just translate, it reshapes. Some regions increase in density while others decrease, all governed by the principle of minimal control effort.
Optimal probability density evolution on SO(2), showing how two nearby peaks are steered and reshaped over time
The circular case is pedagogically valuable because you can visualize everything. You see how the geometry constrains the motion, how the diffusion spreads the distribution, and how control narrows or sharpens it. The same principles apply to SO(3), but visualization becomes harder because you can't easily draw the 3D rotation space.
SO(3): Full 3D rotations
SO(3) is the group of all 3D rotation matrices. Every rotation in 3D corresponds to a unique element of SO(3), up to a small identification. It's a 3-dimensional manifold living naturally in a 9-dimensional Euclidean space, but the manifold itself has its own intrinsic geometry that isn't captured by those ambient coordinates.
Controlling probability distributions on SO(3) is relevant to robotics, computer vision, and any field dealing with orientations or rotations. An inertial measurement unit gives you noisy rotation measurements. A robot needs to coordinate multiple rotating joints. A vision system estimates 3D pose up to uncertainty. All of these naturally live on SO(3), not in Euclidean space.
The main plot in Figure 2 shows an optimal steering problem on SO(3). The evolution from initial to final distribution is smooth and respects the manifold's structure throughout. The computation involved the same iterative refinement seen in SO(2), but now the geometry is more complex. The algorithm handles this automatically, without requiring special treatment for different manifold types.
Connecting to broader research
This work builds on a growing body of research on optimal transport and probability steering on manifolds. The Schrodinger bridge itself has a long history, originally arising in quantum mechanics, but its formulation as an optimal transport problem is more recent. Work on generalized Schrodinger bridges on graphs shows how these ideas extend beyond continuous manifolds. Research on causal Schrodinger bridges and constrained optimal transport explores how causality constraints interact with optimality. And methods for learning non-equilibrium diffusions via Schrodinger bridges show how these frameworks enable data-driven approaches.
The present paper's contribution is to show that all of this can be done without sacrificing the geometry. You don't need to embed or approximate. The coordinate-free approach is not just theoretically cleaner, it makes computation more reliable because you're not accumulating numerical errors from coordinate transformations and constraint projections.
Why this matters
The core insight is simple: if your problem has geometry, respect it. Don't flatten it into Euclidean space and pay the cost in numerical stability, theoretical clarity, and conceptual simplicity.
This principle extends far beyond rotations. Any optimal control problem defined naturally on a compact connected Lie group, now has a clear path forward. The techniques shown here, the existence and uniqueness proofs, the constructive algorithm, all follow from the same principles. The method is templated, waiting to be applied to whatever curved space your problem lives on.
The paper includes working code and animations, publicly available. This matters because it means the results aren't purely theoretical. You can run the algorithm on SO(2) and SO(3) yourself, see the convergence, visualize the optimal steering. Reproducibility like this is how ideas move from papers into practice.
This is a Plain English Papers summary of a research paper called Schrodinger Bridge Over A Compact Connected Lie Group. If you like these kinds of analysis, join AIModels.fyi or follow us on Twitter.