相关论文: Curvature as a tool for evaluating dimensionality …

The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of…

To any metric spaces there is an associated metric profile. The rectifiability of the metric profile gives a good notion of curvature of a sub-Riemannian space. We shall say that a curvature class is the rectifiability class of the metric…

Many applications in network science have recently been discovered for the "curvature" of a network, but there is no consensus on the definition for this term. A common approach in these applications is to derive from the curvature either a…

In order to improve the advantages and the reliability of the second derivative method in tracking the position of extrema from experimental curves, we develop a novel analysis method based on the mathematical concept of curvature. We…

In regression analysis, we employ contour projection (CP) to develop a new dimension reduction theory. Accordingly, we introduce the notions of the central contour subspace and generalized contour subspace. We show that both of their…

We present a novel method for calculating interface curvature on 3D unstructured meshes from piecewise-linear interface reconstructions typically generated in the volume of fluid method. Interface curvature is a necessary quantity to…

In spite of considerable progress, computing curvature in Volume of Fluid (VOF) methods continues to be a challenge. The goal is to develop a function or a subroutine that returns the curvature in computational cells containing an interface…

In this paper we show the similarities and differences of two deep neural networks by comparing the manifolds composed of activation vectors in each fully connected layer of them. The main contribution of this paper includes 1) a new data…

We present a novel hybrid strategy based on machine learning to improve curvature estimation in the level-set method. The proposed inference system couples enhanced neural networks with standard numerical schemes to compute curvature more…

We propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a smooth…

It is a standard assumption that datasets in high dimension have an internal structure which means that they in fact lie on, or near, subsets of a lower dimension. In many instances it is important to understand the real dimension of the…

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several…

Discrete forms of the mean and directed curvature are constructed on piecewise flat manifolds, providing local curvature approximations for smooth manifolds embedded in both Euclidean and non-Euclidean spaces. The resulting expressions take…

Smooth and curved microstructural topologies found in nature - from soap films to trabecular bone - have inspired several mimetic design spaces for architected metamaterials and bio-scaffolds. However, the design approaches so far have been…

The identification of vertices that play a central role in network analysis is a fundamental challenge. Although traditional centrality measures have been extensively employed for this purpose, the increasing complexity of modern networks…

We have recently introduced Forman's discretization of Ricci curvature to the realm of complex networks. Forman curvature is an edge-based measure whose mathematical definition elegantly encapsulates the weights of nodes and edges in a…

The structural analysis of shape boundaries leads to the characterization of objects as well as to the understanding of shape properties. The literature on graphs and networks have contributed to the structural characterization of shapes…

Conforming materials to rigid substrates with Gaussian curvature --- positive for spheres and negative for saddles --- has proven a versatile tool to guide the self-assembly of defects such as scars, pleats, folds, blisters, and liquid…

A plethora of dimensionality reduction techniques have emerged over the past decades, leaving researchers and analysts with a wide variety of choices for reducing their data, all the more so given some techniques come with additional…

During the past two decades there has been a lot of interest in developing statistical depth notions that generalize the univariate concept of ranking to multivariate data. The notion of depth has also been extended to regression models and…