Yongxin Chen

Optimal Steering of a Linear Stochastic System to a Final Probability Distribution—Part III

The subject of this work has its roots in the so-called Schrödginer bridge problem (SBP) which asks for the most likely distribution of Brownian particles in their passage between observed empirical marginal distributions at two distinct points in time. Renewed interest in this problem was sparked by a reformulation in the language of stochastic control. In earlier works, presented as Part I and Part II, we explored a generalization of the original SBP that amounts to optimal steering of linear stochastic dynamical systems between state-distributions, at two points in time, under full state feedback. In these works, the cost was quadratic in the control input, i.e., control energy. The purpose of the present work is to detail the technical steps in extending the framework to the case where a quadratic cost in the state is also present. Thus, the main contribution is to derive the optimal control in this case which in fact is given in closed-form (Theorem 1). In the zero-noise limit, we also obtain the solution of a (deterministic) mass transport problem with general quadratic cost.

On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint

We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou–Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a Fisher information functional. We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior. This can be seen as the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.

Entropic and displacement interpolation: A computational approach using the Hilbert metric

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -- it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for interpolation of distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of recent developments in physics, probability theory, image processing, time-series analysis, and several other fields. In spite of extensive work and theoretical developments, the computation of OMT for large scale problems has remained a challenging task. An alternative framework for interpolating distributions, rooted in statistical mechanics and large deviations, is that of Schroedinger bridges (entropic interpolation). This may be seen as a stochastic regularization of OMT and can be cast as the stochastic control problem of steering the probability density of the state-vector of a dynamical system between two marginals. In this approach, however, the actual computation of flows had hardly received any attention. In recent work on Schroedinger bridges for Markov chains and quantum evolutions, we noted that the solution can be efficiently obtained from the fixed-point of a map which is contractive in the Hilbert metric. Thus, the purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which i) leads to a new proof of a classical result on Schroedinger bridges and ii) provides an efficient computational scheme for both, Schroedinger bridges and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.

Fast cooling for a system of stochastic oscillators

We study feedback control of coupled nonlinear stochastic oscillators in a force field. We first consider the problem of asymptotically driving the system to a desired steady state corresponding to reduced thermal noise. Among the feedback controls achieving the desired asymptotic transfer, we find that the most efficient one from an energy point of view is characterized by time-reversibility. We also extend the theory of Schrodinger bridges to this model, thereby steering the system in finite time and with minimum effort to a target steady-state distribution. The system can then be maintained in this state through the optimal steady-state feedback control. The solution, in the finite-horizon case, involves a space-time harmonic function φ, and −logφ plays the role of an artificial, time-varying potential in which the desired evolution occurs. This framework appears extremely general and flexible and can be viewed as a considerable generalization of existing active control strategies such as macromolecula...

On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches

The positive semidefiniteness of Laplacian matrices corresponding to graphs with negative edge weights is studied. Two alternative proofs to a result by Zelazo and Bürger (Theorem 3.2), which provides upper bounds on the magnitudes of the negative weights in terms of effective resistances within which to ensure definiteness of the Laplacians, are provided. Both proofs are direct and intuitive. The first employs purely geometrical arguments while the second relies on passivity arguments and the laws of physics for electrical circuits. The latter is then used to establish consensus in multi-agent systems with generalized high-order dynamics. A numerical example is given at the end of the paper to highlight the result.

Stochastic Bridges of Linear Systems

We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein-Uhlenbeck process conditioned at the two end-points-a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.

State covariances and the matrix completion problem

State statistics of a linear system obey certain structural constraints that arise from the underlying dynamics and the directionality of input disturbances. Herein, we formulate completion problems of partially known state statistics with the added freedom of identifying disturbance dynamics. The goal of the proposed completion problem is to obtain information about input excitations that explain observed sample statistics. Our formulation aims at low-complexity models for admissible disturbances. The complexity represents the dimensionality of the subspace of the state-dynamics that is directly affected by disturbances. An example is provided to illustrate that colored-in-time stochastic processes can be effectively used to explain available data.

Low-Complexity Modeling of Partially Available Second-Order Statistics: Theory and an Efficient Matrix Completion Algorithm

State statistics of linear systems satisfy certain structural constraints that arise from the underlying dynamics and the directionality of input disturbances. In the present paper, we study the problem of completing partially known state statistics. Our aim is to develop tools that can be used in the context of control-oriented modeling of large-scale dynamical systems. For the type of applications we have in mind, the dynamical interaction between state variables is known while the directionality and dynamics of input excitation is often uncertain. Thus, the goal of the mathematical problem that we formulate is to identify the dynamics and directionality of input excitation in order to explain and complete observed sample statistics. More specifically, we seek to explain correlation data with the least number of possible input disturbance channels. We formulate this inverse problem as rank minimization, and for its solution, we employ a convex relaxation based on the nuclear norm. The resulting optimization problem is cast as a semidefinite program and can be solved using general-purpose solvers. For problem sizes that these solvers cannot handle, we develop a customized alternating minimization algorithm (AMA). We interpret AMA as a proximal gradient for the dual problem and prove sublinear convergence for the algorithm with fixed step-size. We conclude with an example that illustrates the utility of our modeling and optimization framework and draw contrast between AMA and the commonly used alternating direction method of multipliers (ADMM) algorithm.

Optimal steering of inertial particles diffusing anisotropically with losses

Exploiting a fluid dynamic formulation for which a probabilistic counterpart might not be available, we extend the theory of Schrödinger bridges to the case of inertial particles with losses and general, possibly singular diffusion coefficient. We find that, as for the case of constant diffusion coefficient matrix, the optimal control law is obtained by solving a system of two p.d.e.s involving adjoint operators and coupled through their boundary values. In the linear case with quadratic loss function, the system turns into two matrix Riccati equations with coupled split boundary conditions. An alternative formulation of the control problem as a semidefinite programming problem allows computation of suboptimal solutions. This is illustrated in one example of inertial particles subject to a constant rate killing.

Entropic and Displacement Interpolation

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -- it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for interpolation of distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of recent developments in physics, probability theory, image processing, time-series analysis, and several other fields. In spite of extensive work and theoretical developments, the computation of OMT for large scale problems has remained a challenging task. An alternative framework for interpolating distributions, rooted in statistical mechanics and large deviations, is that of Schroedinger bridges (entropic interpolation). This may be seen as a stochastic regularization of OMT and can be cast as the stochastic control problem of steering the probability density of the state-vector of a dynamical system between two marginals. In this approach, however, the actual computation of flows had hardly received any attention. In recent work on Schroedinger bridges for Markov chains and quantum evolutions, we noted that the solution can be efficiently obtained from the fixed-point of a map which is contractive in the Hilbert metric. Thus, the purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which i) leads to a new proof of a classical result on Schroedinger bridges and ii) provides an efficient computational scheme for both, Schroedinger bridges and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.