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

Cellular interactions can be modeled as complex dynamical systems represented by weighted graphs. The functionality of such networks, including measures of robustness, reliability, performance, and efficiency, are intrinsically tied to the topology and geometry of the underlying graph. Utilizing recently proposed geometric notions of curvature on weighted graphs, we investigate the features of gene co-expression networks derived from large-scale genomic studies of cancer. We find that the curvature of these networks reliably distinguishes between cancer and normal samples, with cancer networks exhibiting higher curvature than their normal counterparts. We establish a quantitative relationship between our findings and prior investigations of network entropy. Furthermore, we demonstrate how our approach yields additional, non-trivial pair-wise (i.e. gene-gene) interactions which may be disrupted in cancer samples. The mathematical formulation of our approach yields an exact solution to calculating pair-wise changes in curvature which was computationally infeasible using prior methods. As such, our findings lay the foundation for an analytical approach to studying complex biological networks.

We consider the problem of steering a linear dynamical system with complete state observation from an initial Gaussian distribution in state-space to a final one with minimum energy control. The system is stochastically driven through the control channels; an example for such a system is that of an inertial particle experiencing random "white noise'' forcing. We show that a target probability distribution can always be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitly by solving a pair of differential Lyapunov equations that are coupled through their boundary values. This result, given its attractive algorithmic nature, appears to have several potential applications such as to quality control, industrial processes as well as to active control of nanomechanical systems and molecular cooling. The problem to steer a diffusion process between end-point marginals has a long history (Schroedinger bridges) and therefore, the present case of steering a linear stochastic system constitutes a Schroedinger bridge for possibly degenerate diffusions. Our results, however, provide the first {\em implementable} form of the optimal control for a general Gauss-Markov process. Illustrative examples of the optimal evolution and control for inertial particles and a stochastic oscillator are provided. A final result establishes directly the property of Schroedinger bridges as the most likely random evolution between given marginals to the present context of linear stochastic systems

In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier-Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for colored-intime forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that colored-in-time excitation of the Navier-Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization, and control design.