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Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the…
Shape estimation and object reconstruction are common problems in image analysis. Mathematically, viewing objects in the image plane as random sets reduces the problem of shape estimation to inference about sets. Currently existing…
This article is an exposition on some recent theoretical advances in learning latent structured models, with a primary focus on the fundamental roles that optimal transport distances play in the statistical theory. We aim at what may be the…
Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability…
Transferring a physical system from an initial to a final state while minimizing energetic losses is an interdisciplinary control problem that bridges stochastic thermodynamics and optimal transport theory. Recent research typically…
We propose a methodology to address two analysis problems concerning complex systems, namely bounding state functionals of stochastic differential equations (SDEs) and verifying set avoidance of systems described by partial differential…
Usually, density functional models are considered approximations to density functional theory, However, there is no systematic connection between the two, and this can make us doubt about a linkage. This attitude can be further enforced by…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show…
This paper addresses congested transport, which can be described, at macroscopic scales, by a continuity equation with a pressure variable generated from the hard-congestion constraint (maximum value of the density). The main goal of the…
Motivated by the Swampland Distance Conjecture, we study distances in field space using the framework of Optimal Transport. The associated optimisation problem naturally leads to a notion of distance in terms of a (generalised) Wasserstein…
In sustained growth with random dynamics stationary distributions can exist without detailed balance. This suggests thermodynamical behavior in fast growing complex systems. In order to model such phenomena we apply both a discrete and a…
In this paper we revisit our previous study of the local theory of prescribed Jacobian equations associated with generating functions, which are extensions of cost functions in the theory of optimal transportation. In particular, as…
Time-dependent density functional theory, proposed recently in the context of atomic diffusion and non-equilibrium processes in solids, is tested against Monte Carlo simulation. In order to assess the basic approximation of that theory, the…
We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. This family includes but is not limited to the location of the path supremum and first/last hitting times. We first show that…
This paper is concerned with the study of a class of nonsmooth cost functions subject to a quasi-linear PDE in Lipschitz domains in dimension two. We derive the Eulerian semi-derivative of the cost function by employing the averaged adjoint…
To optimize a neural network one often thinks of optimizing its parameters, but it is ultimately a matter of optimizing the function that maps inputs to outputs. Since a change in the parameters might serve as a poor proxy for the change in…
There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being…
In the geometric transportation problem, we are given a collection of points $P$ in $d$-dimensional Euclidean space, and each point is given a supply of $\mu(p)$ units of mass, where $\mu(p)$ could be a positive or a negative integer, and…
To advance the foundation of one-particle reduced density matrix functional theory (1RDMFT) we refine and relate some of its fundamental features and underlying concepts. We define by concise means the scope of a 1RDMFT, identify its…