arXiv subjects

Michele Pavon

Publications and source records attributed to Michele Pavon.

At least 19 recordsLinked to original sources

arXiv2025

Gradient Flows as Optimal Controlled Evolutions: From Rn to Wasserstein product spaces

We show that the continuous-time gradient descent in Rn can be viewed as an optimal controlled evolution for a suitable action functional; a similar result holds for stochastic gradient descent. We then provide an analogous characterization for the Wasserstein gradient flow of the (relative) entropy, with an action that mirrors the classical case where the Euclidean gradient is replaced by the Wasserstein gradient of the relative entropy. In the small-step limit, these continuous-time actions align with the Jordan Kinderlehrer Otto scheme. Next, we consider gradient flows for the relative entropy over a Wasserstein product space-a study motivated by the stochastic-control formulation of Schrodinger bridges. We characterize the product-space steepest descent as the solution to a variational problem with two control velocities and a product-space Wasserstein gradient, and we show that the induced fluxes in the two components are equal and opposite. This framework suggests applications to the optimal control evolution of microrobotic swarms that can communicate their present distribution to the other swarm.

math.OC↗

arXiv2024

Entropy-regularized optimal transport over networks with incomplete marginals information

We study in this paper optimal mass transport over a strongly connected, directed graph on a given discrete time interval. Differently from previous literature, we do not assume full knowledge of the initial and final goods distribution over the network nodes. In spite of the meager information, we show that it is possible to characterize the most likely flow in two important cases: The first one is when the initial and/or final distribution is only known on proper subsets of the nodes. The second case is when only some moments of the marginal distributions are known. As an important by-product, we determine the most likely initial and final marginals on the whole state space.

math.PR↗

arXiv2023

On the rate of change of Varentropy for Markov diffusion processes

Relying on the reverse-time space-time harmonic property of the ratio of two solutions of the Fokker-Plank equation, we establish an explicit formula for derivative of the \emph{varentropy} for a Markov diffusion process. The formula involves a nonlinear function of the {\em local free energy} $\ln(p_t(x)/\bar{p}(x))$.We then verify that our formula yields the correct result in the simple case of a scalar Gaussian diffusion. In the latter case, varentropy is exponentially decaying to zero.

math.PR↗

arXiv2023

Control and estimation of multi-commodity network flow under aggregation

A paradigm put forth by E. Schrödinger in 1931/32, known as Schrödinger 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 Schrödinger-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 Schrödinger'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 Schrödinger 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.

math.OC↗

arXiv2022

On local entropy, stochastic control and deep neural networks

In this paper, we connect some recent papers on smoothing of energy landscapes and scored-based generative models of machine learning to classical work in stochastic control. We clarify these connections providing rigorous statements and representations which may serve as guidelines for further learning models.

math.OC↗

arXiv2021

Harmonic Analysis of Social Cognition

In this paper, we argue that some fundamental concepts and tools of signal processing may be effectively applied to represent and interpret social cognition processes. From this viewpoint, individuals or, more generally, social stimuli are thought of as a weighted sum of harmonics with different frequencies: Low frequencies represent general categories such as gender, ethnic group, nationality, etc., whereas high frequencies account for personal characteristics. Individuals are then seen by observers as the output of a filter that emphasizes a certain range of high or low frequencies. The selection of the filter depends on the social distance between the observing individual or group and the person being observed as well as on motivation, cognitive resources and cultural background. Enhancing low- or high-frequency harmonics is not on equal footing, the latter requiring supplementary energy. This mirrors a well-known property of signal processing filters. More generally, in the light of this correspondence, we show that several established results of social cognition admit a natural interpretation and integration in the signal processing language. While the potential of this connection between an area of social psychology and one of information engineering appears considerable (compression, information retrieval, filtering, feedback, feedforward, sampling, aliasing, etc.), in this paper we shall limit ourselves to laying down what we consider the pillars of this bridge on which future research may be founded.

math.SP↗

arXiv2021

The most likely evolution of diffusing and vanishing particles: Schrodinger Bridges with unbalanced marginals

Stochastic flows of an advective-diffusive nature are ubiquitous in physical sciences. Of particular interest is the problem to reconcile observed marginal distributions with a given prior posed by E. Schrodinger in 1932/32 and known as the Schrodinger Bridge Problem (SBP). Due to its fundamental significance, interest in SBP has in recent years enticed a broad spectrum of disciplines. Yet, while the mathematics and applications of SBP have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention; the problem to interpolate between unbalanced marginals has been approached by introducing source/sink terms in an Adhoc manner. Nevertheless, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schrodinger's dictum; that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated law represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for stochastic evolution with losses, whereupon particles are "killed" according to a probabilistic law. Through a suitable embedding, we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism, given a prior law that allows for losses, we ask for the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman-Kac multiplicative reweighing of the reference measure: The latter, as we argue, is far from Schrodinger's quest.

math-ph↗

arXiv2021

Optimal steering to invariant distributions for networks flows

We derive novel results on the ergodic theory of irreducible, aperiodic Markov chains. We show how to optimally steer the network flow to a stationary distribution over a finite or infinite time horizon. Optimality is with respect to an entropic distance between distributions on feasible paths. When the prior is reversible, it shown that solutions to this discrete time and space steering problem are reversible as well. A notion of temperature is defined for Boltzmann distributions on networks, and problems analogous to cooling (in this case, for evolutions in discrete space and time) are discussed.

eess.SY↗

arXiv2020

Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge

In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of large deviations of the empirical distribution and an early example of the so-called maximum entropy inference method. This so-called Schroedinger bridge problem (SBP) was recently recognized as a regularization of the Monge-Kantorovich Optimal Mass Transport (OMT), leading to effective computation of the latter. Specifically, OMT with quadratic cost may be viewed as a zero-temperature limit of SBP, which amounts to minimization of the Helmholtz's free energy over probability distributions constrained to possess given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called {\em Sinkhorn algorithm} which appears as a special case of an algorithm first studied by the French analyst Robert Fortet in 1938/40 specifically for Schroedinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation by iteratively solving the so-called {\em Schroedinger system}. In both the continuous as well as the discrete-time and space settings, {\em stochastic control} provides a reformulation and dynamic versions of these problems. The formalism behind these control problems have attracted attention as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, network routing as well as in computer and data science. This multifacet and versatile framework, intertwining SBP and OMT, provides the substrate for a historical and technical overview of the field taken up in this paper.

math.OC↗

arXiv2020

Regularized transport between singular covariance matrices

We consider the problem of steering a linear stochastic system between two end-point degenerate Gaussian distributions in finite time. This accounts for those situations in which some but not all of the state entries are uncertain at the initial, t = 0, and final time, t = T . This problem entails non-trivial technical challenges as the singularity of terminal state-covariance causes the control to grow unbounded at the final time T. Consequently, the entropic interpolation (Schroedinger Bridge) is provided by a diffusion process which is not finite-energy, thereby placing this case outside of most of the current theory. In this paper, we show that a feasible interpolation can be derived as a limiting case of earlier results for non-degenerate cases, and that it can be expressed in closed form. Moreover, we show that such interpolation belongs to the same reciprocal class of the uncontrolled evolution. By doing so we also highlight a time-symmetry of the problem, contrasting dual formulations in the forward and reverse time-directions, where in each the control grows unbounded as time approaches the end-point (in the forward and reverse time-direction, respectively).

math.OC↗

arXiv2019

Covariance steering in zero-sum linear-quadratic two-player differential games

We formulate a new class of two-person zero-sum differential games, in a stochastic setting, where a specification on a target terminal state distribution is imposed on the players. We address such added specification by introducing incentives to the game that guides the players to steer the join distribution accordingly. In the present paper, we only address linear quadratic games with Gaussian target distribution. The solution is characterized by a coupled Riccati equations system, resembling that in the standard linear quadratic differential games. Indeed, once the incentive function is calculated, our problem reduces to a standard one. Tthe framework developed in this paper extends previous results in covariance control, a fast growing research area. On the numerical side, problems herein are reformulated as convex-concave minimax problems for which efficient and reliable algorithms are available.

eess.SY↗

arXiv2019

Controlled and Uncontrolled Stochastic Norton-Simon-Massagué Tumor Growth Models

Tumorigenesis is a complex process that is heterogeneous and affected by numerous sources of variability. This study presents a stochastic extension of a biologically grounded tumor growth model, referred to as the Norton-Simon-Massagué (NSM) tumor growth model. We first study the uncontrolled version of the model where the effect of chemotherapeutic drug agent is absent. Conditions on the model's parameters are derived to guarantee the positivity of the tumor volume and hence the validity of the proposed stochastic NSM model. To calibrate the proposed model we utilize a maximum likelihood-based estimation algorithm and population mixed-effect modeling formulation. The algorithm is tested by fitting previously published tumor volume mice data. Then, we study the controlled version of the model which includes the effect of chemotherapy treatment. Analysis of the influence of adding the control drug agent into the model and how sensitive it is to the stochastic parameters is performed both in open-loop and closed-loop viewpoints through different numerical simulations.

q-bio.TO↗

arXiv2019

Optimal steering for non-Markovian Gaussian processes

At present, the problem to steer a non-Markovian process with minimum energy between specified end-point marginal distributions remains unsolved. Herein, we consider the special case for a non-Markovian process y(t) which, however, assumes a finite-dimensional stochastic realization with a Markov state process that is fully observable. In this setting, and over a finite time horizon [0,T], we determine an optimal (least) finite-energy control law that steers the stochastic system to a final distribution that is compatible with a specified distribution for the terminal output process y(T); the solution is given in closed-form. This work provides a key step towards the important problem to steer a stochastic system based on partial observations of the state (i.e., an output process) corrupted by noise, which will be the subject of forthcoming work.

math.OC↗

arXiv2018

Traversing the Schroedinger Bridge strait: Robert Fortet's marvelous proof redux

In the early 1930's, Erwin Schroedinger, motivated by his quest for a more classical formulation of quantum mechanics, posed a large deviation problem for a cloud of independent Brownian particles. He showed that the solution to the problem could be obtained trough a system of two linear equations with nonlinear coupling at the boundary (Schrödinger system). Existence and uniqueness for such a system, which represents a sort of bottleneck for the problem, was first established by R. Fortet in 1938/40 under rather general assumptions by proving convergence of an ingenious but complex approximation method. It is the first proof of what are nowadays called Sinkhorn-type algorithms in the much more challenging continuous case. Schrödinger bridges are also an early example of the maximum entropy approach and have been more recently recognized as a regularization of the important Optimal Mass Transport problem. Unfortunately, Fortet's contribution is by and large ignored in contemporary literature. This is likely due to the complexity of his approach coupled with an idiosyncratic exposition style and to missing details and steps in the proofs. Nevertheless, Fortet's approach maintains its importance to this day as it provides the only existing algorithmic proof under rather mild assumptions. It can be adapted, in principle, to other relevant problems such as the regularized Wasserstein barycenter problem. It is the purpose of this paper to remedy this situation by rewriting the bulk of his paper with all the missing passages and in a transparent fashion so as to make it fully available to the scientific community. We consider the problem in $R^d$ rather than $R$ and use as much as possible his notation to facilitate comparison.

math.OC↗

arXiv2018

The data-driven Schroedinger bridge

Erwin Schroedinger posed, and to a large extent solved in 1931/32 the problem of finding the most likely random evolution between two continuous probability distributions. This article considers this problem in the case when only samples of the two distributions are available. A novel iterative procedure is proposed, inspired by Fortet-Sinkhorn type algorithms. Since only samples of the marginals are available, the new approach features constrained maximum likelihood estimation in place of the nonlinear boundary couplings, and importance sampling to propagate the functions $φ$ and $\hatφ$ solving the Schroedinger system. This method is well-suited to high-dimensional settings, where introducing grids leads to numerically unfeasible or unreliable methods. The methodology is illustrated in two applications: entropic interpolation of two-dimensional Gaussian mixtures, and the estimation of integrals through a variation of importance sampling.

math.OC↗

arXiv2018

Ruelle-Bowen continuous-time random walk

We define the probability structure of a continuous-time time-homogeneous Markov jump process, on a finite graph, that represents the continuous-time counterpart of the so-called Ruelle-Bowen discrete-time random walk. It constitutes the unique jump process having maximal entropy rate. Moreover, it has the property that, given the number of jumps between any two specified end-points on the graph, the probability of traversing any one of the alternative paths that are consistent with the specified number of jumps and end-points, is the same for all, and thereby depends only on the number of jumps and the end-points and not the particular path being traversed.

math.OC↗

arXiv2018

Relaxed Schroedinger bridges and robust network routing

We consider network routing under random link failures with a desired final distribution. We provide a mathematical formulation of a relaxed transport problem where the final distribution only needs to be close to the desired one. The problem is a maximum entropy problem for path distributions with an extra terminal cost. We show that the unique solution may be obtained solving a generalized Schroedinger system. An iterative algorithm to compute the solution is provided. It contracts the Hilbert metric with contraction ratio less than 1/2 leading to extremely fast convergence.

math.OC↗

arXiv2017

Steering the distribution of agents in mean-field and cooperative games

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.

eess.SY↗