Related papers: Random projections of random manifolds

We consider the discrete three dimensional scan statistics. Viewed as the maximum of an 1-dependent stationary r.v.'s sequence, we provide approximations and error bounds for the probability distribution of the three dimensional scan…

We consider the problem of reconstructing the intrinsic geometry of a manifold from noisy pairwise distance observations. Specifically, let $M$ denote a diameter 1 d-dimensional manifold and $\mu$ a probability measure on $M$ that is…

Monocular depth prediction plays a crucial role in understanding 3D scene geometry. Although recent methods have achieved impressive progress in terms of evaluation metrics such as the pixel-wise relative error, most methods neglect the…

Given a probability distribution P, what is the minimum amount of bits needed to store a value x sampled according to P, such that x can later be recovered (except with some small probability)? Or, what is the maximum amount of uniform…

We introduce abstract rendering, a method for computing a set of images by rendering a scene from a continuously varying range of camera positions. The resulting abstract image-which encodes an infinite collection of possible renderings-is…

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are…

High-dimensional data are ubiquitous, with examples ranging from natural images to scientific datasets, and often reside near low-dimensional manifolds. Leveraging this geometric structure is vital for downstream tasks, including signal…

We study the limit geometry of complete projective special real manifolds. By limit geometry we mean the limit of the evolution of the defining polynomial and the centro-affine fundamental form along certain curves that leave every compact…

We give explicit theoretical and heuristical bounds for how big does a data set sampled from a reach-1 submanifold M of euclidian space need to be, to be able to estimate the dimension of M with 90% confidence.

This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a…

Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$…

Quantum mechanics is sensitive to the geometry of the underlying space. Here, we present a framework for quantum scattering of a non-relativistic particle confined to a two-dimensional space. When the motion manifold hosts localized…

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with…

Recently, multiple formulations of vision problems as probabilistic inversions of generative models based on computer graphics have been proposed. However, applications to 3D perception from natural images have focused on low-dimensional…

While classical data analysis has addressed observations that are real numbers or elements of a real vector space, at present many statistical problems of high interest in the sciences address the analysis of data that consist of more…

Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step…

Sectional curvature bounds are of central importance in the study of Riemannian manifolds, both in smooth differential geometry and in the generalized synthetic setting of Alexandrov spaces. Riemannian metrics along with metric spaces of…

Exploratory Landscape Analysis (ELA) provides numerical features for characterizing black-box optimization problems. In high-dimensional settings, however, ELA suffers from sparsity effects, high estimator variance, and the prohibitive cost…

The manifold hypothesis suggests that high-dimensional data often lie on or near a low-dimensional manifold. Estimating the dimension of this manifold is essential for leveraging its structure, yet existing work on dimension estimation is…

Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in…