Distribution Steering via Sliced Optimal Transport Control

Abstract: Distribution steering seeks feedback laws that drive the state law of a dynamical system between prescribed initial and terminal distributions. Optimal transport provides a natural geometric approach, but its implementation generally requires a transport map or coupling in the full state space. Sliced optimal transport avoids this full-dimensional construction through one-dimensional projections. Yet, the resulting projected maps specify only directional displacements and do not by themselves prescribe a realizable feedback law. To this end, we develop a finite-horizon control framework based on sliced optimal transport. At each sampling instant, a projected optimal transport map defines a directional terminal condition, whose minimum-energy realization yields a randomized single-direction controller. Averaging over projection directions gives a deterministic sliced feedback. For the single-integrator dynamics, the averaged feedback makes the sliced Wasserstein distance to the target non-increasing. For Gaussian endpoint laws, it is affine, preserves Gaussianity, and steers the mean and covariance to their prescribed terminal values. We further identify a law-dependent gain that yields linear decay of the sliced Wasserstein distance together with an explicit characterization of the control energy. We also prove that the randomized controller converges to the averaged sliced flow as the sampling period vanishes. Finally, we extend the construction to linear dynamical systems. Reachability-normalized coordinates allow instantaneous realization of the sliced velocity for uniformly fully actuated systems, while local controllability Gramians provide exact finite-step realization for general controllable systems. Numerical examples illustrate the resulting distributional flows.

• The paper develops a feedback controller using one-dimensional optimal transport maps over random projections, proving monotone sliced-Wasserstein descent and terminal convergence for Gaussian distributions.
• The paper characterizes the energy cost of sliced control, showing an overhead bounded by a factor of n divided by the directional-consistency coefficient and achieving 1.31 times exact OT energy in an image-transfer benchmark.
• The paper proves that single-direction randomized updates converge with expected squared Wasserstein error O(h), while extending the method to fully actuated and general controllable linear systems through coordinate transformations and finite-step displacement matching.

Motivation and problem statement

Distribution steering asks for a feedback law that drives the state law $\rho_t$ of a controlled system from an initial distribution $\rho_0$ to a target distribution $\rho_1$. Classical optimal transport (OT)-based controllers, including the Benamou–Brenier dynamic formulation and the linear-quadratic constructions of Chen, Georgiou, and Pavon, require a transport map or coupling in the full ambient space $\mathbb{R}^n$. Constructing and repeatedly recomputing such an object is expensive for large sample sets. The paper develops an alternative: a finite-horizon feedback framework built exclusively from one-dimensional optimal transport maps between linear projections of the current and target laws, i.e., sliced optimal transport. The central gap the authors identify is that a projected transport map specifies only a directional displacement and does not by itself prescribe a realizable feedback law; the contribution is to close this gap by treating projected maps as directional terminal conditions whose minimum-energy realizations generate feedback.

The sliced controller and its averaged flow

For the single integrator $\dot{x} = u$, at each sampling instant $t_k$ a direction $\theta_k$ is drawn and the scalar endpoint problem of driving $\theta_k^\top x_k$ to its transported image $T_k^{\theta_k}(\theta_k^\top x_k)$ at the terminal time is solved in minimum energy. The receding-horizon lift yields

$$u_k = -\lambda(t_k)\left(\theta_k^\top x_k - T_k^{\theta_k}(\theta_k^\top x_k)\right)\theta_k,$$

where $\rho_0$0 recovers the remaining-horizon gain. Averaging over $\rho_0$1 (uniform on the sphere) produces the deterministic averaged sliced feedback $\rho_0$2, where $\rho_0$3 is the sliced discrepancy field. The paper's first structural result (Proposition 1) shows that $\rho_0$4 is locally absolutely continuous along any sufficiently regular continuity-equation flow and that

$\rho_0$5

Hence the averaged feedback is a descent flow for the sliced Wasserstein distance. A directional-consistency coefficient $\rho_0$6 quantifies how much of the total sliced discrepancy survives directional averaging; values near one indicate reinforcement, small values indicate cancellation, and the inequality $\rho_0$7 follows from Cauchy–Schwarz.

Gaussian specialization

When both endpoint laws are Gaussian, one-dimensional OT maps are affine, and Proposition 2 shows the averaged sliced feedback is affine, $\rho_0$8, preserves Gaussianity, and reduces the dynamics to closed-form mean and covariance equations. The mean evolves as $\rho_0$9, while the covariance obeys a matrix ODE driven by $\rho_1$0. Theorem 1 establishes terminal convergence: under a divergent cumulative gain $\rho_1$1 and a uniform lower bound on the eigenvalues of $\rho_1$2, the sliced distance decays exponentially,

$\rho_1$3

with an explicit, dimension-dependent constant $\rho_1$4, and $\rho_1$5. Notably, since the same affine feedback governs the first two moments of any law with finite second moment, mean-and-covariance steering holds even for non-Gaussian laws.

Energy characterization

Lemma 1 provides the paper's sharpest quantitative statement: for any averaged sliced flow, the instantaneous control power satisfies

$\rho_1$6

separating the decay rate from the averaging loss. A law-dependent gain $\rho_1$7 then yields exact linear decay $\rho_1$8 (Theorem 2), with total energy

$\rho_1$9

bounded between $\mathbb{R}^n$0 and $\mathbb{R}^n$1. Since the classical minimum energy equals $\mathbb{R}^n$2, the averaged sliced controller incurs an energy overhead of at most a factor $\mathbb{R}^n$3 — a concrete price for avoiding full-dimensional transport. The Gaussian corollary strengthens this by proving $\mathbb{R}^n$4 whenever $\mathbb{R}^n$5, so the gain is well defined and the covariance stays uniformly positive definite.

Convergence of the randomized controller

The implementable controller uses a single random direction per step, and Theorem 3 shows it is a consistent stochastic Euler discretization of the averaged flow: under an existence assumption for the averaged flow and an $\mathbb{R}^n$6-stability assumption on the sliced field along generated laws,

$\mathbb{R}^n$7

The proof decomposes each update into the averaged drift plus a martingale-difference fluctuation (bounded via uniform moment estimates and preservation of absolute continuity of the pushforward maps) and applies a discrete Gronwall argument. The $\mathbb{R}^n$8 expected squared Wasserstein error establishes that the randomized, single-direction scheme is not merely heuristic but converges to the analyzed feedback. The assumptions are stated as assumptions — well-posedness of the averaged flow and stability of $\mathbb{R}^n$9 are not proved in general, only verified in the Gaussian case.

Extensions to linear dynamics

For linear time-varying systems $\dot{x} = u$0 with $\dot{x} = u$1 of full row rank (uniformly fully actuated), the drift is removed by $\dot{x} = u$2 and control authority is normalized by the reachability Gramian via $\dot{x} = u$3 with $\dot{x} = u$4. The minimum-norm control $\dot{x} = u$5 then realizes the sliced velocity exactly in normalized coordinates, transferring terminal convergence to the original system (Proposition 3). Physical input energy is equivalent to the transformed kinetic energy up to the actuation bounds $\dot{x} = u$6:

$\dot{x} = u$7

For general controllable systems where $\dot{x} = u$8 may be rank deficient, instantaneous velocity matching fails; the paper instead performs finite-step displacement matching: over each interval of a partition, the input $\dot{x} = u$9 realizes the sliced displacement exactly by the interval end (Proposition 4), so the law at sampling instants follows the virtual sliced iteration with no additional discretization error. The authors are explicit that this construction is intrinsically tied to the finite partition: the local Gramian $t_k$0 can become increasingly ill-conditioned as intervals shrink, so refining the partition does not generally yield a finite-energy continuous-time limit.

Numerical evidence

Two experiments illustrate the framework. A Gaussian-mixture steering problem in $t_k$1 with $t_k$2 particles shows monotone decay of the empirical sliced Wasserstein distance consistent with the dissipation identity, with terminal discrepancy comparable to the sampling noise between two independent target samples. A color-transfer experiment (Monet's Haystack toward Water Lilies) realizes the fully actuated controller through a time-varying linear system in CIELAB color space, reducing the sliced distance from 0.299 to 0.066 while preserving spatial structure; the measured input energy remains within the predicted $t_k$3 bounds. As an energy benchmark on a 1500-pixel subsample, the sliced controller uses 1.31 times the exact OT minimum energy. Computationally, the sliced update costs $t_k$4 per step versus $t_k$5 memory and cubic worst-case time for exact assignment: one exact map at $t_k$6 takes 15.3 s, while a complete 60-step sliced run at full image resolution $t_k$7 takes 1.1 s with roughly 5.2 MB of storage versus approximately 3.4 GB for the dense cost matrix.

Limitations and open questions

Several caveats are stated in the paper itself. The convergence theory for the randomized controller rests on Assumptions 1 and 2 (existence of the averaged flow and $t_k$8-stability of the sliced field), which are verified only for Gaussian laws; general well-posedness of the averaged sliced flow remains open. The energy bounds depend on a uniform lower bound on the directional-consistency coefficient $t_k$9, which is established only in the Gaussian case, and the linear-decay gain $\theta_k$0 is law-dependent and requires knowledge of the current distribution. The finite-step realization for general controllable systems is exact only at sampling instants and admits no continuous-time limit when $\theta_k$1 is rank deficient. The construction also presumes full state feedback and known endpoint laws; adaptive projection selection, partial observations, and stochastic, constrained, or nonlinear dynamics are explicitly left as open directions.

Conclusion

This paper converts sliced optimal transport from a discrepancy measure into a dynamically realizable feedback methodology. Its main results — monotone sliced-Wasserstein descent, Gaussian-preserving affine feedback with terminal convergence, an explicit energy characterization with a bounded optimality gap, vanishing-step convergence of the randomized controller, and exact realizations for fully actuated and general controllable linear systems — collectively demonstrate that projected one-dimensional transport maps suffice for finite-horizon distribution steering at a computational cost that scales to large sample sets. The framework's guarantees are currently sharpest for Gaussian laws and fully actuated dynamics, and extending them beyond these regimes is the principal unresolved question.