Related papers: Baby Morse Theory in Data Analysis
Topological Data Analysis (TDA) can broadly be described as a collection of data analysis methods that find structure in data. This includes: clustering, manifold estimation, nonlinear dimension reduction, mode estimation, ridge estimation…
Topological Data Analysis (TDA) has recently gained significant attention in the field of financial prediction. However, the choice of point cloud construction methods, topological feature representations, and classification models has a…
A point cloud serves as a representation of the surface of a three-dimensional (3D) shape. Deep generative models have been adapted to model their variations typically using a map from a ball-like set of latent variables. However, previous…
We use the nudged elastic band method from computational chemistry to analyze high-dimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…
We present a way to apply topological data analysis for classifying encrypted bits into distinct classes. Persistent homology is applied to generate topological features of a point cloud obtained from sets of encryptions. We see that this…
Advanced 3D metrology technologies such as Coordinate Measuring Machine (CMM) and laser 3D scanners have facilitated the collection of massive point cloud data, beneficial for process monitoring, control and optimization. However, due to…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
We develop a general framework for the probabilistic analysis of random finite point clouds in the context of topological data analysis. We extend the notion of a barcode of a finite point cloud to compact metric spaces. Such a barcode…
We develop a language for describing the relationship among observations, mathematical models, and the underlying principles from which they are derived. Using Information Geometry, we consider geometric properties of statistical models for…
Processing point clouds using deep neural networks is still a challenging task. Most existing models focus on object detection and registration with deep neural networks using point clouds. In this paper, we propose a deep model that learns…
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation…
In this paper, we introduce a novel method for comparing 3D point clouds, a critical task in various machine learning applications. By interpreting point clouds as samples from underlying probability density functions, the statistical…
Topology at the undergraduate level is often a theoretical mathematics course, introducing concepts from point-set topology or possibly algebraic topology. However, the last two decades have seen an explosion of growth in applied topology…
Topological Data Analysis (TDA) is a novel, and relatively new approach to analysing high-dimensional data sets. It does this by focussing on global properties like the shape and connectivity of the data giving it a significant advantage…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar-valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to…
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…
Point cloud synthesis, i.e. the generation of novel point clouds from an input distribution, remains a challenging task, for which numerous complex machine learning models have been devised. We develop a novel method that encodes…
This paper focuses on a novel generative approach for 3D point clouds that makes use of invertible flow-based models. The main idea of the method is to treat a point cloud as a probability density in 3D space that is modeled using a…