相关论文: A geometric framework for asymptotic inference of …

We study all four-dimensional simply-connected indecomposable non-semisimple pseudo-Riemannian symmetric spaces whose metric has signature (2,2). We present models and compute their isometry groups. We solve the problem of the existence or…

Motivated by a paper of Zirnbauer, we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the…

Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches…

The asymptotic results pertaining to the distribution of the log likelihood ratio allow for the creation of a confidence region, which is a general extension of the confidence interval. Two and three dimensional regions can be displayed…

Asymptotic efficiency theory is one of the pillars in the foundations of modern mathematical statistics. Not only does it serve as a rigorous theoretical benchmark for evaluating statistical methods, but it also sheds light on how to…

We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible…

By making use of the symplectic reduction and the cohomogeneity method, we give a general method for constructing Hamiltonian minimal submanifolds in Kaehler manifolds with symmetries. As applications, we construct infinitely many…

Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High…

We treat the notion of principal symbol mapping on a compact smooth manifold as a $\ast$-homomorphism of $C^{\ast}$-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the…

We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric…

Identifying low-dimensional latent structures within high-dimensional data has long been a central topic in the machine learning community, driven by the need for data compression, storage, transmission, and deeper data understanding.…

This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…

A large number of algorithms in machine learning, from principal component analysis (PCA), and its non-linear (kernel) extensions, to more recent spectral embedding and support estimation methods, rely on estimating a linear subspace from…

We prove an integral formula for the spherical measure of hypersurfaces in equiregular sub-Riemannian manifolds. Among various technical tools, we establish a general criterion for the uniform convergence of parametrized sub-Riemannian…

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to…

We propose a methodology for constructing confidence regions with partially identified models of general form. The region is obtained by inverting a test of internal consistency of the econometric structure. We develop a dilation bootstrap…

We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution.…