A Generalized Sinkhorn Algorithm for Mean-Field Schrödinger Bridge

The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.

Generative Profiling for Soft Real-Time Systems and its Applications to Resource Allocation

Modern real-time systems require accurate characterization of task timing behavior to ensure predictable performance, particularly on complex hardware architectures. Existing methods, such as worst-case execution time analysis, often fail to capture the fine-grained timing behaviors of a task under varying resource contexts (e.g., an allocation of cache, memory bandwidth, and CPU frequency), which is necessary to achieve efficient resource utilization. In this paper, we introduce a novel generative profiling approach that synthesizes context-dependent, fine-grained timing profiles for real-time tasks, including those for unmeasured resource allocations. Our approach leverages a nonparametric, conditional multi-marginal Schrödinger Bridge (MSB) formulation to generate accurate execution profiles for unseen resource contexts, with maximum likelihood guarantees. We demonstrate the efficiency and effectiveness of our approach through real-world benchmarks, and showcase its practical utility in a representative case study of adaptive multicore resource allocation for real-time systems.

Symmetrizing Bregman Divergence on the Cone of Positive Definite Matrices: Which Mean to Use and Why

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

Schrödinger Bridge Over A Compact Connected Lie Group

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 $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$.

Set Invariance with Probability One for Controlled Diffusion: Score-based Approach

Given a controlled diffusion and a connected, bounded, Lipschitz set, when is it possible to guarantee controlled set invariance with probability one? In this work, we answer this question by deriving the necessary and sufficient conditions for the same in terms of gradients of certain log-likelihoods -- a.k.a. score vector fields -- for two cases: given finite time horizon and infinite time horizon. The deduced conditions comprise a score-based test that provably certifies or falsifies the existence of Markovian controllers for given controlled set invariance problem data. Our results are constructive in the sense when the problem data passes the proposed test, we characterize all controllers guaranteeing the desired set invariance. When the problem data fails the proposed test, there does not exist a controller that can accomplish the desired set invariance with probability one. The computation in the proposed tests involve solving certain Dirichlet boundary value problems, and in the finite horizon case, can also account for additional constraint of hitting a target subset at the terminal time. We illustrate the results using several semi-analytical and numerical examples.

Markov Kernels, Distances and Optimal Control: A Parable of Linear Quadratic Non-Gaussian Distribution Steering

For a controllable linear time-varying (LTV) pair $(\boldsymbol{A}_t,\boldsymbol{B}_t)$ and $\boldsymbol{Q}_{t}$ positive semidefinite, we derive the Markov kernel for the It\^{o} diffusion ${\mathrm{d}}\boldsymbol{x}_{t}=\boldsymbol{A}_{t}\boldsymbol{x}_t {\mathrm{d}} t + \sqrt{2}\boldsymbol{B}_{t}{\mathrm{d}}\boldsymbol{w}_{t}$ with an accompanying killing of probability mass at rate $\frac{1}{2}\boldsymbol{x}^{\top}\boldsymbol{Q}_{t}\boldsymbol{x}$. This Markov kernel is the Green's function for an associated linear reaction-advection-diffusion partial differential equation. Our result generalizes the recently derived kernel for the special case $\left(\boldsymbol{A}_t,\boldsymbol{B}_t\right)=\left(\boldsymbol{0},\boldsymbol{I}\right)$, and depends on the solution of an associated Riccati matrix ODE. A consequence of this result is that the linear quadratic non-Gaussian Schr\"{o}dinger bridge is exactly solvable. This means that the problem of steering a controlled LTV diffusion from a given non-Gaussian distribution to another over a fixed deadline while minimizing an expected quadratic cost can be solved using dynamic Sinkhorn recursions performed with the derived kernel. Our derivation for the $\left(\boldsymbol{A}_t,\boldsymbol{B}_t,\boldsymbol{Q}_t\right)$-parametrized kernel pursues a new idea that relies on finding a state-time dependent distance-like functional given by the solution of a deterministic optimal control problem. This technique breaks away from existing methods, such as generalizing Hermite polynomials or Weyl calculus, which have seen limited success in the reaction-diffusion context. Our technique uncovers a new connection between Markov kernels, distances, and optimal control. This connection is of interest beyond its immediate application in solving the linear quadratic Schr\"{o}dinger bridge problem.

The Ground Cost for Optimal Transport of Angular Velocity

We revisit the optimal transport problem over angular velocity dynamics given by the controlled Euler equation. The solution of this problem enables stochastic guidance of spin states of a rigid body (e.g., spacecraft) over hard deadline constraint by transferring a given initial state statistics to a desired terminal state statistics. This is an instance of generalized optimal transport over a nonlinear dynamical system. While prior work has reported existence-uniqueness and numerical solution of this dynamical optimal transport problem, here we present structural results about the equivalent Kantorovich a.k.a. optimal coupling formulation. Specifically, we focus on deriving the ground cost for the associated Kantorovich optimal coupling formulation. The ground cost equals to the cost of transporting unit amount of mass from a specific realization of the initial or source joint probability measure to a realization of the terminal or target joint probability measure, and determines the Kantorovich formulation. Finding the ground cost leads to solving a structured deterministic nonlinear optimal control problem, which is shown to be amenable to an analysis technique pioneered by Athans et. al. We show that such techniques have broader applicability in determining the ground cost (thus Kantorovich formulation) for a class of generalized optimal mass transport problems involving nonlinear dynamics with translated norm-invariant drift.

Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps

For a given ground cost, approximating the Monge optimal transport map that pushes forward a given probability measure onto another has become a staple in several modern machine learning algorithms. The fourth-order Ma-Trudinger-Wang (MTW) tensor associated with this ground cost function provides a notion of curvature in optimal transport. The non-negativity of this tensor plays a crucial role for establishing continuity for the Monge optimal transport map. It is, however, generally difficult to analytically verify this condition for any given ground cost. To expand the class of cost functions for which MTW non-negativity can be verified, we propose a provably correct computational approach which provides certificates of non-negativity for the MTW tensor using Sum-of-Squares (SOS) programming. We further show that our SOS technique can also be used to compute an inner approximation of the region where MTW non-negativity holds. We apply our proposed SOS programming method to several practical ground cost functions to approximate the regions of regularity of their corresponding optimal transport maps.

Weyl Calculus and Exactly Solvable Schrödinger Bridges with Quadratic State Cost

Schr\"{o}dinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schr\"{o}dinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schr\"{o}dinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schr\"{o}dinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials.

Schrödinger Bridge with Quadratic State Cost is Exactly Solvable

Schr\"odinger bridge is a diffusion process that steers a given distribution to another in a prescribed time while minimizing the effort to do so. It can be seen as the stochastic dynamical version of the optimal mass transport, and has growing applications in generative diffusion models and stochastic optimal control. In this work, we propose a regularized variant of the Schr\"odinger bridge with a quadratic state cost-to-go that incentivizes the optimal sample paths to stay close to a nominal level. Unlike the conventional Schr\"odinger bridge, the regularization induces a state-dependent rate of killing and creation of probability mass, and its solution requires determining the Markov kernel of a reaction-diffusion partial differential equation. We derive this Markov kernel in closed form. Our solution recovers the heat kernel in the vanishing regularization (i.e., diffusion without reaction) limit, thereby recovering the solution of the conventional Schr\"odinger bridge. Our results enable the use of dynamic Sinkhorn recursion for computing the Schr\"odinger bridge with a quadratic state cost-to-go, which would otherwise be challenging to use in this setting. We deduce properties of the new kernel and explain its connections with certain exactly solvable models in quantum mechanics.