Related papers: Betti Number for Point Sets

Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…

Alternative set theory (AST) may be suitable for the ones who try to capture objects or phenomenons with some kind of indefiniteness of a border. While AST provides various notions for advanced mathematical studies, correspondence of them…

In many applications, it is needed to change the topology of a tensor network directly and without approximation. This work will introduce a general scheme that satisfies these needs. We will describe the procedure by two examples and show…

We present a detail-driven deep neural network for point set upsampling. A high-resolution point set is essential for point-based rendering and surface reconstruction. Inspired by the recent success of neural image super-resolution…

Persistence diagrams serve as a core tool in topological data analysis, playing a crucial role in pathological monitoring, drug discovery, and materials design. However, existing quantum topological algorithms, such as the LGZ algorithm,…

Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti numbers are given.

In topological quantum computing, information is encoded in "knotted" quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by…

Persistent homology is one of the most active branches of Computational Algebraic Topology with applications in several contexts such as optical character recognition or analysis of point cloud data. In this paper, we report on the formal…

Meshes are widely used in 3D computer vision and graphics, but their irregular topology poses challenges in applying them to existing neural network architectures. Recent advances in mesh neural networks turn to remeshing and push the…

The aim of this article is to present a topological tool for the study of additive basis in additive number theory. It will be proposal a metric for the set of all additive basis, in which it will be possible to study properties of some…

Machine learning models for repeated measurements are limited. Using topological data analysis (TDA), we present a classifier for repeated measurements which samples from the data space and builds a network graph based on the data topology.…

Persistent homology is a fundamental tool in topological data analysis; however, it lacks methods to quantify the fragility or fineness of cycles, anticipate their formation or disappearance, or evaluate their stability beyond persistence.…

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of…

The Bott index has become an indispensable tool to probe the topology of quantum matter, particularly in systems lacking translational symmetry. Constructed from a plaquette operator, it retains the phase information while discarding the…

The ideals generated by fold products of linear forms are generalizations of powers of defining ideals of star configurations, or of Veronese type ideals, and in this paper we study their Betti numbers. In earlier work, the authors together…

We study how the topology of feature embedding space changes as it passes through the layers of a well-trained deep neural network (DNN) through Betti numbers. Motivated by existing studies using simplicial complexes on shallow fully…

Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of…

Swept volume computation, the determination of regions occupied by moving objects, is essential in graphics, robotics, and manufacturing. Existing approaches either explicitly track surfaces, suffering from robustness issues under complex…

Computer-Aided Design is ubiquitous in todays world, as almost every manufactured object begins as a digital model across industries. At the same time, advances in 3D sensing have made point clouds a dominant form of raw 3D data. Recovering…

We introduce a level set based approach to Bayesian geometric inverse problems. In these problems the interface between different domains is the key unknown, and is realized as the level set of a function. This function itself becomes the…