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Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample…
The manifold of empirical mean values of statistical data ad infinitum has a geometric shape that depends on the probability measure that governs the generating model. Large deviation theory produces entropy functions that depend on both…
This paper deals with the problem of quantifying the approximation a probability measure by means of an empirical (in a wide sense) random probability measure, depending on the first n terms of a sequence of random elements. In Section 2,…
Metrics for rigorously defining a distance between two events have been used to study the properties of the dataspace manifold of particle collider physics. The probability distribution of pairwise distances on this dataspace is unique with…
A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their…
Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their…
We present a technique, which we call "etching," which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4. The harmonic measure is the probability distribution of random walkers diffusing…
In this paper, we study rare events in spherical and Gaussian random geometric graphs in high dimensions. In these models, the vertices correspond to points sampled uniformly at random on the $d$ dimensional unit sphere or correspond to $d$…
Ensembles of random stochastic and bistochastic matrices are investigated. While all columns of a random stochastic matrix can be chosen independently, the rows and columns of a bistochastic matrix have to be correlated. We evaluate the…
The experimental problem of converting a measured binomial quantity, the fraction of events in a sample that pass a cut, into a physical binomial quantity, the fraction of events originating from a signal source, is described as a system of…
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs.…
Random geometric graphs (RGGs) are commonly used to model networked systems that depend on the underlying spatial embedding. We concern ourselves with the probability distribution of an RGG, which is crucial for studying its random…
We propose a new metric between probability measures on a compact metric space that mirrors the Riemannian manifold-like structure of quadratic optimal transport but includes entropic regularization. Its metric tensor is given by the…
The approximation of probability measures on compact metric spaces and in particular on Riemannian manifoldsby atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of…
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint…
We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad…
Given a probability measure $\mu$ on a set $\mathcal{X}$ and a vector-valued function $\varphi$, a common problem is to construct a discrete probability measure on $\mathcal{X}$ such that the push-forward of these two probability measures…
We develop a notion of projections between sets of probability measures using the geometric properties of the 2-Wasserstein space. It is designed for general multivariate probability measures, is computationally efficient to implement, and…
This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular $\el$-sided polygon. Using this result, we obtain the closed-form probability…
We introduce probability-graphons which are probability kernels that generalize graphons to the case of weighted graphs. Probability-graphons appear as the limit objects to study sequences of large weighted graphs whose distribution of…