Related papers: Algebraic and Topological Persistence
We study the foundational properties of persistent homotopy groups and develop elementary computational methods for their analysis. Our main theorems are persistent analogues of the Van Kampen, excision, suspension, and Hurewicz theorems.…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
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…
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…
We consider simplicial sets equipped with a notion of smallness, and observe that this slight "topological" extension of the "algebraic" simplicial language allows a concise reformulation of a number of classical notions in topology, e.g.…
One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…
The \v{C}ech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech filtration, the number of simplices grows exponentially in the number of input points. A…
Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a…
The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes,…
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…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We…
A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and…
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a…
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in…
Recently, it was found that there is a remarkable intuitive similarity between studies in theoretical computer science dealing with large data sets on the one hand, and categorical methods of topology and geometry in pure mathematics, on…
Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring…