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In this paper, three Computational Topology methods (namely effective homology, persistent homology and discrete vector fields) are mixed together to produce algorithms for homological digital image processing. The algorithms have been…
We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…
There has been an intense recent activity in embedding of very high dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part we cover…
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…
Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…
Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale…
We present a data-driven framework to automate the vectorization and machine interpretation of 2D engineering part drawings. In industrial settings, most manufacturing engineers still rely on manual reads to identify the topological and…
Node2vec is a graph embedding method that learns a vector representation for each node of a weighted graph while seeking to preserve relative proximity and global structure. Numerical experiments suggest Node2vec struggles to recreate the…
Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$ that can…
We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with…
We apply persistent homology, the main method in topological data analysis, to the study of demographic data. Persistence diagrams efficiently summarize information about clusters or peaks in a region's demographic data. To illustrate how…
Data analysis that uses the output of topological data analysis as input for machine learning algorithms has been the subject of extensive research. This approach offers a means of capturing the global structure of data. Persistent homology…
We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes: digital, position, and interaction, motivated by different capabilities of…
The machine learning technique of persistent homology classifies complex systems or datasets by computing their topological features over a range of characteristic scales. There is growing interest in applying persistent homology to…
The capability of accurate prediction of protein functions and properties is essential in the biotechnology industry, e.g. drug development and artificial protein synthesis, etc. The main challenges of protein function prediction are the…
Many multi-variate time series obtained in the natural sciences and engineering possess a repetitive behavior, as for instance state-space trajectories of industrial machines in discrete automation. Recovering the times of recurrence from…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…
Despite progress in the rapidly developing field of geometric deep learning, performing statistical analysis on geometric data--where each observation is a shape such as a curve, graph, or surface--remains challenging due to the…
The aim of this paper is to investigate an attempt to build a binary classification algorithm using principles of geometry such as vectors, planes, and vector algebra. The basic idea behind the proposed algorithm is that a hyperplane can be…
Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here we develop a mathematical pipeline that systematically analyzes protein structures. We showcase this…