43 research outputs found
Control and estimation of multi-commodity network flow under aggregation
A paradigm put forth by E. Schr\"odinger in 1931/32, known as Schr\"odinger
bridges, represents a formalism to pose and solve control and estimation
problems seeking a perturbation from an initial control schedule (in the case
of control), or from a prior probability law (in the case of estimation),
sufficient to reconcile data in the form of marginal distributions and minimal
in the sense of relative entropy to the prior. In the same spirit, we consider
traffic-flow and apply a Schr\"odinger-type dictum, to perturb minimally with
respect to a suitable relative entropy functional a prior schedule/law so as to
reconcile the traffic flow with scarce aggregate distributions on families of
indistinguishable individuals. Specifically, we consider the problem to
regulate/estimate multi-commodity network flow rates based only on empirical
distributions of commodities being transported (e.g., types of vehicles through
a network, in motion) at two given times. Thus, building on Schr\"odinger's
large deviation rationale, we develop a method to identify {\em the most likely
flow rates (traffic flow)}, given prior information and aggregate observations.
Our method further extends the Schr\"odinger bridge formalism to the
multi-commodity setting, allowing commodities to exit or enter the flow field
as well (e.g., vehicles to enter and stop and park) at any time. The behavior
of entering or exiting the flow field, by commodities or vehicles, is modeled
by a Markov chains with killing and creation states. Our method is illustrated
with a numerical experiment.Comment: 12 pages, 5 figure
Provably Convergent Schr\"odinger Bridge with Applications to Probabilistic Time Series Imputation
The Schr\"odinger bridge problem (SBP) is gaining increasing attention in
generative modeling and showing promising potential even in comparison with the
score-based generative models (SGMs). SBP can be interpreted as an
entropy-regularized optimal transport problem, which conducts projections onto
every other marginal alternatingly. However, in practice, only approximated
projections are accessible and their convergence is not well understood. To
fill this gap, we present a first convergence analysis of the Schr\"odinger
bridge algorithm based on approximated projections. As for its practical
applications, we apply SBP to probabilistic time series imputation by
generating missing values conditioned on observed data. We show that optimizing
the transport cost improves the performance and the proposed algorithm achieves
the state-of-the-art result in healthcare and environmental data while
exhibiting the advantage of exploring both temporal and feature patterns in
probabilistic time series imputation.Comment: Accepted by ICML 202
A kernel-based method for Schr\"odinger bridges
We characterize the Schr\"odinger bridge problems by a family of
Mckean-Vlasov stochastic control problems with no terminal time distribution
constraint. In doing so, we use the theory of Hilbert space embeddings of
probability measures and then describe the constraint as penalty terms defined
by the maximum mean discrepancy in the control problems. A sequence of the
probability laws of the state processes resulting from -optimal
controls converges to a unique solution of the Schr\"odinger's problem under
mild conditions on given initial and terminal time distributions and an
underlying diffusion process. We propose a neural SDE based deep learning
algorithm for the Mckean-Vlasov stochastic control problems. Several numerical
experiments validate our methods
Steering the Distribution of Agents in Mean-Field Games System
Abstract The purpose of this work is to pose and solve the problem to guide a collection of weakly interacting dynamical systems (agents, particles, etc.) to a specified terminal distribution. The framework is that of mean-field and of cooperative games. A terminal cost is used to accomplish the task; we establish that the map between terminal costs and terminal probability distributions is onto. Our approach relies on and extends the theory of optimal mass transport and its generalizations
Wasserstein Hamiltonian flow with common noise on graph
We study the Wasserstein Hamiltonian flow with a common noise on the density
manifold of a finite graph. Under the framework of stochastic variational
principle, we first develop the formulation of stochastic Wasserstein
Hamiltonian flow and show the local existence of a unique solution. We also
establish a sufficient condition for the global existence of the solution.
Consequently, we obtain the global well-posedness for the nonlinear
Schr\"odinger equations with common noise on graph. In addition, using
Wong-Zakai approximation of common noise, we prove the existence of the
minimizer for an optimal control problem with common noise. We show that its
minimizer satisfies the stochastic Wasserstein Hamiltonian flow on graph as
well
On the Contraction Coefficient of the Schr\"odinger Bridge for Stochastic Linear Systems
Schr\"{o}dinger bridge is a stochastic optimal control problem to steer a
given initial state density to another, subject to controlled diffusion and
deadline constraints. A popular method to numerically solve the Schr\"{o}dinger
bridge problems, in both classical and in the linear system settings, is via
contractive fixed point recursions. These recursions can be seen as dynamic
versions of the well-known Sinkhorn iterations, and under mild assumptions,
they solve the so-called Schr\"{o}dinger systems with guaranteed linear
convergence. In this work, we study a priori estimates for the contraction
coefficients associated with the convergence of respective Schr\"{o}dinger
systems. We provide new geometric and control-theoretic interpretations for the
same. Building on these newfound interpretations, we point out the possibility
of improved computation for the worst-case contraction coefficients of linear
SBPs by preconditioning the endpoint support sets
Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances
We develop structure preserving schemes for a class of nonlinear mobility
continuity equation. When the mobility is a concave function, this equation
admits a form of gradient flow with respect to a Wasserstein-like transport
metric. Our numerical schemes build upon such formulation and utilize modern
large scale optimization algorithms. There are two distinctive features of our
approach compared to previous ones. On one hand, the essential properties of
the solution, including positivity, global bounds, mass conservation and energy
dissipation are all guaranteed by construction. On the other hand, it enjoys
sufficient flexibility when applies to a large variety of problems including
different free energy functionals, general wetting boundary conditions and
degenerate mobilities. The performance of our methods are demonstrated through
a suite of examples.Comment: 24 pages, 12 figure
Extremal flows in Wasserstein space
We develop an intrinsic geometric approach to the calculus of variations in theWasserstein
space. We show that the flows associated with the Schr\ua8odinger bridge with
general prior, with optimal mass transport, and with the Madelung fluid can all be
characterized as annihilating the first variation of a suitable action. We then discuss
the implications of this unified framework for stochastic mechanics: It entails, in particular,
a sort of fluid-dynamic reconciliation between Bohm\u2019s and Nelson\u2019s stochastic
mechanics