Related papers: Curvature of point clouds through principal compon…

Principal Component Analysis (PCA) is a workhorse of modern data science. While PCA assumes the data conforms to Euclidean geometry, for specific data types, such as hierarchical and cyclic data structures, other spaces are more…

Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering…

A kernel based method is proposed for the construction of signature (defining) functions of subsets of $\mathbb{R}^d$. The subsets can range from full dimensional manifolds (open subsets) to point clouds (a finite number of points) and…

Existing learning-based point cloud upsampling methods often overlook the intrinsic data distribution charac?teristics of point clouds, leading to suboptimal results when handling sparse and non-uniform point clouds. We propose a novel…

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if…

We consider the evolution of starshaped hypersurfaces in the Euclidean space by general curvature functions. Under appropriate conditions on the curvature function, we prove the global existence and convergence of the flow to a hypersurface…

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably,…

In machine learning it is common to interpret each data point as a vector in Euclidean space. However the data may actually be functional i.e.\ each data point is a function of some variable such as time and the function is discretely…

Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This paper investigates…

We propose a family of curvature-based regularization terms for deep generative model learning. Explicit coordinate-invariant formulas for both intrinsic and extrinsic curvature measures are derived for the case of arbitrary data manifolds…

Determining the representativeness of a point within a data cloud has recently become a desirable task in multivariate analysis. The concept of statistical depth function, which reflects centrality of an arbitrary point, appears to be…

The success of algorithms in the analysis of high-dimensional data is often attributed to the manifold hypothesis, which supposes that this data lie on or near a manifold of much lower dimension. It is often useful to determine or estimate…

One develops a fast computational methodology for principal component analysis on manifolds. Instead of estimating intrinsic principal components on an object space with a Riemannian structure, one embeds the object space in a numerical…

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…

In this work, we develop new generalization bounds for neural networks trained on data supported on Riemannian manifolds. Existing generalization theories often rely on complexity measures derived from Euclidean geometry, which fail to…

Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic, or…

In this paper, we propose a point cloud classification method based on graph neural network and manifold learning. Different from the conventional point cloud analysis methods, this paper uses manifold learning algorithms to embed point…

3D building models with facade details are playing an important role in many applications now. Classifying point clouds at facade-level is key to create such digital replicas of the real world. However, few studies have focused on such…

Functional data analysis deals with data recorded densely over time (or any other continuum) with one or more observed curves per subject. Conceptually, functional data are continuously defined, but in practice, they are usually observed at…

We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of…