Related papers: Sequents, barcodes, and homology
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
Graph Signal Processing deals with the problem of analyzing and processing signals defined on graphs. In this paper, we introduce a novel filtering method for graph-based signals by employing ideas from topological data analysis. We begin…
Sequence theories are an extension of theories of strings with an infinite alphabet of letters, together with a corresponding alphabet theory (e.g. linear integer arithmetic). Sequences are natural abstractions of extendable arrays, which…
We derive the relationship between the persistent homology barcodes of two dual filtered CW complexes. Applied to greyscale digital images, we obtain an algorithm to convert barcodes between the two different (dual) topological models of…
We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis,…
Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a ubiquitous tool in TDA. Basing our study on this, we…
We introduce harmonic persistent homology spaces for filtrations of finite simplicial complexes. As a result we can associate concrete subspaces of cycles to each bar of the barcode of the filtration. We prove stability of the harmonic…
Lineage tracing, the determination and mapping of progeny arising from single cells, is an important approach enabling the elucidation of mechanisms underlying diverse biological processes ranging from development to disease. We developed a…
In medical image analysis, feature engineering plays an important role in the design and performance of machine learning models. Persistent homology (PH), from the field of topological data analysis (TDA), demonstrates robustness and…
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…
New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from…
We define a simple, explicit map sending a morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this…
Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their…
Intending to introduce a method for the topological analysis of fields, we present a pipeline that takes as an input a weighted and based chain complex, produces a factored chain complex, and encodes it as a barcode of tagged intervals…
Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking topological features across changes in different scales. Classical algorithms for persistent homology are often constrained…
While there has been much interest in adapting conventional clustering procedures---and in higher dimensions, persistent homology methods---to directed networks, little is known about the convergence of such methods. In order to even…
We introduce Reeb complexes in order to capture how generators of homology flow along sections of a real valued continuous function. This intuition suggests a close relation of Reeb complexes to established methods in topological data…
This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…
Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze…
We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of…