Related papers: Persistence Norms and the Datasaurus
Topological Data Analysis (TDA), an emerging field in investment sciences, harnesses mathematical methods to extract data features based on shape, offering a promising alternative to classical portfolio selection methodologies. We utilize…
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes…
Recently many efforts have been made to incorporate persistence diagrams, one of the major tools in topological data analysis (TDA), into machine learning pipelines. To better understand the power and limitation of persistence diagrams, we…
Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…
Persistent homology allows us to create topological summaries of complex data. In order to analyse these statistically, we need to choose a topological summary and a relevant metric space in which this topological summary exists. While…
One of the primary areas of interest in applied algebraic topology is persistent homology, and, more specifically, the persistence diagram. Persistence diagrams have also become objects of interest in topological data analysis. However,…
Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable,…
This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. A new distance on…
Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis -…
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in…
What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient…
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…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
Persistence diagrams are two-dimensional plots that summarize the topological features of functions and are an important part of topological data analysis. A problem that has received much attention is how deal with sets of persistence…
Using a set of $\Lambda$CDM simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art tools of multiscale topological data analysis (TDA). We…
Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA --…
The stability of persistent homology has led to wide applications of the persistence diagram as a trusted topological descriptor in the presence of noise. However, with the increasing demand for high-dimension and low-sample-size data…
Topological Data Analysis (TDA) is a field that leverages tools and ideas from algebraic topology to provide robust methods for analysing geometric and topological aspects of data. One of the principal tools of TDA, persistent homology,…
Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the study of the shape of the data. In this work we investigate the predictive power of TDA in the context of supervised learning. Since topological…
Latent space matching, which consists of matching distributions of features in latent space, is a crucial component for tasks such as adversarial attacks and defenses, domain adaptation, and generative modelling. Metrics for probability…