Related papers: A roadmap for the computation of persistent homolo…
Persistent topological Laplacians are operators that provide persistent Betti numbers and additional multiscale geometric information through the eigenvalues of the persistent topological Laplacian matrix. We introduce a framework and novel…
Persistent homology has been devised as a promising tool for the topological simplification of complex data. However, it is computationally intractable for large data sets. In this work, we introduce multiresolution persistent homology for…
Persistent topological properties of an image serve as an additional descriptor providing an insight that might not be discovered by traditional neural networks. The existing research in this area focuses primarily on efficiently…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
In this paper, we introduce the persistence transformation, a novel methodology in Topological Data Analysis (TDA) for applications in time series data which can be obtained in various areas such as science, politics, economy, healthcare,…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to…
Existing interpretability methods for Large Language Models (LLMs) predominantly capture linear directions or isolated features. This overlooks the high-dimensional, relational, and nonlinear geometry of model representations. We apply…
Visualization in the emerging field of topological data analysis has progressed from persistence barcodes and persistence diagrams to display of two-parameter persistent homology. Although persistence barcodes and diagrams have permitted…
As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically…
We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in…
Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics and transport is one of most challenging tasks in biological science. In the present work, persistent homology is,…
Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…
Persistent homology computes the multiscale topology of a data set by using a sequence of discrete complexes. In this paper, we propose that persistent homology may be a useful tool for studying the structure of the landscape of string…
0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…
Topological Data Analysis (TDA) is a modern approach to Data Analysis focusing on the topological features of data; it has been widely studied in recent years and used extensively in Biology, Physics, and many other areas. However,…
In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure:…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…
Topological data analysis (TDA) has emerged as one of the most promising techniques to reconstruct the unknown shapes of high-dimensional spaces from observed data samples. TDA, thus, yields key shape descriptors in the form of persistent…
Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to…