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We present a new fast approximate algorithm for Tukey (halfspace) depth level sets and its implementation-ABCDepth. Given a $d$-dimensional data set for any $d\geq 1$, the algorithm is based on a representation of level sets as…
We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite…
An usual problem in statistics consists in estimating the minimizer of a convex function. When we have to deal with large samples taking values in high dimensional spaces, stochastic gradient algorithms and their averaged versions are…
A method to approximate continuous multi-dimensional probability density functions (PDFs) using their projections and correlations is described. The method is particularly useful for event classification when estimates of systematic…
We design an efficient data structure for computing a suitably defined approximate depth of any query point in the arrangement $\mathcal{A}(S)$ of a collection $S$ of $n$ halfplanes or triangles in the plane or of halfspaces or simplices in…
Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time…
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…
Space-filling experimental designs are widely used in engineering computer experiments, where only a limited number of expensive model evaluations can be afforded. Distance-based designs such as Maximin or Minimax ensure global…
Projective splitting is a family of methods for solving inclusions involving sums of maximal monotone operators. First introduced by Eckstein and Svaiter in 2008, these methods have enjoyed significant innovation in recent years, becoming…
We consider the geometry relaxation of an isolated point defect embedded in a homogeneous crystalline solid, within an atomistic description. We prove a sharp convergence rate for a periodic supercell approximation with respect to uniform…
We establish some results for the rate of convergence in total variation of a Gibbs sampler to its equilibrium distribution. This sampler is motivated by a hierarchical Bayesian inference construction for a gamma random variable. Our…
Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to…
Data depths are score functions that quantify in an unsupervised fashion how central is a point inside a distribution, with numerous applications such as anomaly detection, multivariate or functional data analysis, arising across various…
The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
Self-supervised surround-view depth estimation enables dense, low-cost 3D perception with a 360{\deg} field of view from multiple minimally overlapping images. Yet, most existing methods suffer from depth estimates that are inconsistent…
Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of…
In this paper, we show that the halfspace depth random variable for samples from a univariate distribution with a notion of center is distributed as a uniform distribution on the interval [0,1/2]. The simplicial depth random variable has a…
This work constructs Jonson-Lindenstrauss embeddings with best accuracy, as measured by variance, mean-squared error and exponential concentration of the length distortion. Lower bounds for any data and embedding dimensions are determined,…
We establish efficient approximate counting algorithms for several natural problems in local lemma regimes. In particular, we consider the probability of intersection of events and the dimension of intersection of subspaces. Our approach is…