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The optimal transportation problem defines a geometry of probability measures which leads to a definition for weighted averages (barycenters) of measures, finding application in the machine learning and computer vision communities as a…
The second part of the paper mainly deals with convergence of infinite determinantal measures, understood as the convergence of the approximating finite determinantal measures. In addition to the usual weak topology on the space of…
We study finite probability theory through a category of finite probability schemes and probability-preserving maps, called \emph{bundles}. A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and…
In many applications, the two-dimensional trajectories of fluid particles are available, but little is known about the underlying flow. Oceanic floats are a clear example. To extract quantitative information from such data, one can measure…
The paper investigates the approximation of the symmetric Total Variation functional on graphs. Such an approximation is given in terms of a discrete and symmetric finite difference model defined on point clouds obtained by randomly…
This work presents a new method to quantify connectivity in transportation networks. Inspired by the field of topological data analysis, we propose a novel approach to explore the robustness of road network connectivity in the presence of…
Galaxy surveys provide finite catalogs of objects observed within bounded volumes, yet clustering statistics are often interpreted using theoretical frameworks developed for infinite point processes. In this work, we formulate key…
Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from,…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
Let C(K) be the Banach space of all continuous functions on a given compact space K. We investigate the w*-sequential closure in C(K)* of the set of all finitely supported probabilities on K. We discuss the coincidence of the Baire…
Probabilistic databases (PDBs) are used to model uncertainty in data in a quantitative way. In the standard formal framework, PDBs are finite probability spaces over relational database instances. It has been argued convincingly that this…
We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere…
We generalize the definition of topological entropy given by Adler, Konheim, and McAndrew (AKM) for piecewise continuous self-maps defined on a compact interval (pc-maps). For this notion of entropy, we prove that the properties of the…
We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a…
Providing generalization guarantees for stochastic optimization algorithms remains a key challenge in learning theory. Recently, numerous works demonstrated the impact of the geometric properties of optimization trajectories on…
We introduce the post-processing preorder and equivalence relations for general measurements on a possibly infinite-dimensional general probabilistic theory described by an order unit Banach space $E$ with a Banach predual. We define the…
The concept of $typed$ $topology$ is introduced. In a typed topological space, some open sets are assigned "types", and topological concepts such as closure, connectedness can be defined using types. A finite data set in $R^2$ is a…
In this paper, we propose a unified framework for sampling, clustering and embedding data points in semi-metric spaces. For a set of data points $\Omega=\{x_1, x_2, \ldots, x_n\}$ in a semi-metric space, we consider a complete graph with…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
This review aims to cover the central aspects of current research in cosmic topology from a topological and observational perspective. Beginning with an overview of the basic concepts of cosmology, it is observed that though a determinant…