Related papers: Distribution of Euclidean Distances Between Random…
We consider an \eps-approximation by n-term partial sums of the Karhunen-Lo\`eve expansion to d-parametric random fields of tensor product-type in the average case setting. We investigate the behavior, as d tends to infinity, of the…
Gaussian processes are a widely embraced technique for regression and classification due to their good prediction accuracy, analytical tractability and built-in capabilities for uncertainty quantification. However, they suffer from the…
Clustering is a crucial task in various domains of knowledge, including medicine, epidemiology, genomics, environmental science, economics, and visual sciences, among others. Methodologies for inferring the number of clusters have often…
Bayesian variable selection has gained much empirical success recently in a variety of applications when the number $K$ of explanatory variables $(x_1,...,x_K)$ is possibly much larger than the sample size $n$. For generalized linear…
Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using…
Studies on various facets of pattern classification is often imperative while working with multi-dimensional samples pertaining to diverse application scenarios. In this notion, weighted dimension-based distance measure has been one of the…
We tackle the problem of trajectory planning in an environment comprised of a set of obstacles with uncertain time-varying locations. The uncertainties are modeled using widely accepted Gaussian distributions, resulting in a…
While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in…
We develop a projected Wasserstein distance for the two-sample test, a fundamental problem in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. In particular, we aim to…
If a line cuts randomly two sides of a triangle, the length of the segment determined by the points of intersection is also random. The object of this study, applied to a particular case, is to calculate the probability that the length of…
Let $G_n$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X_1,\ldots,X_n$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $\|X_i -X_j\| \le r_n$, for a given threshold…
This paper introduces a new clustering technique, called {\em dimensional clustering}, which clusters each data point by its latent {\em pointwise dimension}, which is a measure of the dimensionality of the data set local to that point.…
High-dimensional computational challenges are frequently explained via the curse of dimensionality, i.e., increasing the number of dimensions leads to exponentially growing computational complexity. In this commentary, we argue that…
We examine rules for predicting whether a point in $\mathbb{R}$ generated from a 50-50 mixture of two different probability distributions came from one distribution or the other, given limited (or no) information on the two distributions,…
Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…
In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision…
Unbiased random vectors i.e. distributed uniformly in n-dimensional space, are widely applied and the computational cost of generating a vector increases only linearly with n. On the other hand, generating uniformly distributed random…
High dimensional data analysis is known to be as a challenging problem. In this article, we give a theoretical analysis of high dimensional classification of Gaussian data which relies on a geometrical analysis of the error measure. It…
This paper establishes the theoretical foundation for statistical applications of an intriguing new type of spatial point processes called critical point processes. These point processes, residing in Euclidean space, consist of the critical…
Clustering is an important exploratory data analysis technique to group objects based on their similarity. The widely used $K$-means clustering method relies on some notion of distance to partition data into a fewer number of groups. In the…