Related papers: Experiments with the Census
In recent years various results about locally symmetric manifolds were proven using probabilistic approaches. One of the approaches is to consider random manifolds by associating a probability measure to the space of discrete subgroups of…
For data living in a manifold $M\subseteq \mathbb{R}^m$ and a point $p\in M$ we consider a statistic $U_{k,n}$ which estimates the variance of the angle between pairs of vectors $X_i-p$ and $X_j-p$, for data points $X_i$, $X_j$, near $p$,…
Small area population estimates are useful for decision making in the private and public sectors. However, in small areas (i.e., those that are difficult to reach and with small population sizes) computing demographic quantities is…
We show that in a broad class of random counting measures one may identify only three that are rescaled versions of themselves when restricted to a subspace. These are Poisson, binomial and negative binomial random measures. We provide some…
Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction -- how to assemble discrete samples into global…
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial…
We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…
An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the…
Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…
In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. For this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose…
We propose a statistical framework to investigate whether a given subpopulation lies between two other subpopulations in a multivariate feature space. This methodology is motivated by a biological question from a collaborator: Is a newly…
Two-sample tests are important areas aiming to determine whether two collections of observations follow the same distribution or not. We propose two-sample tests based on integral probability metric (IPM) for high-dimensional samples…
The distribution function of a random distance in three dimensions is given and some new three-dimensional d2-tests of randomness are suggested. We show that our test statistics are not correlated with the usual test statistics and are…
Population size estimation based on capture-recapture experiment under triple record system is an interesting problem in various fields including epidemiology, population studies, etc. In many real life scenarios, there exists inherent…
Motivated by population studies of Diffusion Tensor Imaging, the paper investigates the use of mean-based and dispersion-based permutation tests to define and compute the significance of a statistical test for data taking values on…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
The "large p, small n" paradigm arises in microarray studies, where expression levels of thousands of genes are monitored for a small number of subjects. There has been an increasing demand for study of asymptotics for the various…
The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are…
This paper uses sample data to study the problem of comparing populations on finite-dimensional parallelizable Riemannian manifolds and more general trivial vector bundles. Utilizing triviality, our framework represents populations as…
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping…