Related papers: Computing the Expected Value and Variance of Geome…
For a probability measure on a real separable Hilbert space, we are interested in "volume-based" approximations of the d-dimensional least squares error of it, i.e., least squares error with respect to a best fit d-dimensional affine…
In this paper, a statistically optimal solution to the Perspective-n-Point (PnP) problem is presented. Many solutions to the PnP problem are geometrically optimal, but do not consider the uncertainties of the observations. In addition, it…
We consider a facility location problem, where the objective is to ``disperse'' a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected…
We develop a new formulation of Stein's method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a…
We use the Sum of Squares method to develop new efficient algorithms for learning well-separated mixtures of Gaussians and robust mean estimation, both in high dimensions, that substantially improve upon the statistical guarantees achieved…
Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…
We want to approximate general multivariate probability density functions by deterministic sample sets. For optimal sampling, the closeness to the given continuous density has to be assessed. This is a difficult challenge in multivariate…
Many real objects are modeled as discrete sets of points, such as corners or other salient features. For our main applications in chemistry, points represent atomic centers in a molecule or a solid material. We study the problem of…
A Riemannian geometric framework for Markov chain Monte Carlo (MCMC) is developed where using the Fisher-Rao metric on the manifold of probability density functions (pdfs), informed proposal densities for Metropolis-Hastings (MH) algorithms…
Given two distributions $P$ and $S$ of equal total mass, the Earth Mover's Distance measures the cost of transforming one distribution into the other, where the cost of moving a unit of mass is equal to the distance over which it is moved.…
Non-reversible Markov chain Monte Carlo schemes based on piecewise deterministic Markov processes have been recently introduced in applied probability, automatic control, physics and statistics. Although these algorithms demonstrate…
Random forests are considered one of the best out-of-the-box classification and regression algorithms due to their high level of predictive performance with relatively little tuning. Pairwise proximities can be computed from a trained…
We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a…
Motivated by the problem of compressing point sets into as few bits as possible while maintaining information about approximate distances between points, we construct random nonlinear maps $\varphi_\ell$ that compress point sets in the…
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…
We consider the problem of determining the optimal block (or subsample) size for a spatial subsampling method for spatial processes observed on regular grids. We derive expansions for the mean square error of the subsampling variance…
This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…
K-Means clustering algorithm is one of the most commonly used clustering algorithms because of its simplicity and efficiency. K-Means clustering algorithm based on Euclidean distance only pays attention to the linear distance between…
Predictive geometric models deliver excellent results for many Machine Learning use cases. Despite their undoubted performance, neural predictive algorithms can show unexpected degrees of instability and variance, particularly when applied…
A central part of geometric statistics is to compute the Fr\'echet mean. This is a well-known intrinsic mean on a Riemannian manifold that minimizes the sum of squared Riemannian distances from the mean point to all other data points. The…