Related papers: Curvature as a tool for evaluating dimensionality …
Text documents are complex high dimensional objects. To effectively visualize such data it is important to reduce its dimensionality and visualize the low dimensional embedding as a 2-D or 3-D scatter plot. In this paper we explore…
We review the notion of shape tensor of an embedded manifold, which efficiently combines intrinsic and extrinsic geometry, and allows for intuitive understanding of some basic concepts of classical differential geometry, such as parallel…
We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-C\'ardenas and Gibou, J. Comput. Phys. (May 2022),…
Majority of the current dimensionality reduction or retrieval techniques rely on embedding the learned feature representations onto a computable metric space. Once the learned features are mapped, a distance metric aids the bridging of gaps…
The contemporary scientific landscape is characterized by a "curse of dimensionality," where our capacity to collect high-dimensional network data frequently outstrips our ability to computationally simulate or intuitively comprehend the…
Bump hunting deals with finding in sample spaces meaningful data subsets known as bumps. These have traditionally been conceived as modal or concave regions in the graph of the underlying density function. We define an abstract bump…
This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a…
Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of…
Hyperbolic neural networks (HNNs) have demonstrated notable efficacy in representing real-world data with hierarchical structures via exploiting the geometric properties of hyperbolic spaces characterized by negative curvatures. Curvature…
We propose a novel approach to sufficient dimension reduction in regression, based on estimating contour directions of negligible variation for the response surface. These directions span the orthogonal complement of the minimal space…
This thesis investigates low-dimensional models of nonperturbative quantum gravity, with a special focus on Causal Dynamical Triangulations (CDT). We define the so-called curvature profile, a new quantum gravitational observable based on…
In this paper, we propose an efficient method to estimate the Weingarten map for point cloud data sampled from manifold embedded in Euclidean space. A statistical model is established to analyze the asymptotic property of the estimator. In…
Deep metric learning aims to learn a function mapping image pixels to embedding feature vectors that model the similarity between images. Two major applications of metric learning are content-based image retrieval and face verification. For…
Graph generative model evaluation necessitates understanding differences between graphs on the distributional level. This entails being able to harness salient attributes of graphs in an efficient manner. Curvature constitutes one such…
Man-made objects usually exhibit descriptive curved features (i.e., curve networks). The curve network of an object conveys its high-level geometric and topological structure. We present a framework for extracting feature curve networks…
Dimensionality reduction (DR) on the manifold includes effective methods which project the data from an implicit relational space onto a vectorial space. Regardless of the achievements in this area, these algorithms suffer from the lack of…
We propose a novel measure of statistical depth, the metric spatial depth, for data residing in an arbitrary metric space. The measure assigns high (low) values for points located near (far away from) the bulk of the data distribution,…
Real-world datasets are often of high dimension and effected by the curse of dimensionality. This hinders their comprehensibility and interpretability. To reduce the complexity feature selection aims to identify features that are crucial to…
In the process of projecting the surface of a three-dimensional object onto a two-dimensional surface, due to the perspective distortion, the image on the surface of the object will have different degrees of distortion according to the…
We present a novel algorithm to compute multi-scale curvature fields on triangle meshes. Our algorithm is based on finding robust mean curvatures using the ball neighborhood, where the radius of a ball corresponds to the scale of the…