Related papers: Bipath Persistence as Zigzag Persistence
In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology,…
One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the…
Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical…
Recently, bipath persistent homology has been proposed as an extension of standard persistent homology, along with its visualization (bipath persistence diagram) and computational methods. In the setting of standard persistent homology, the…
Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…
The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
We show that every infinite zigzag persistence module decomposes into a direct sum of interval persistence modules.
The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of…
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…
Algebraic persistence studies persistence modules (typically, linear representations of the poset $\mathbf{R}^n$ with $n \geq 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of…
We apply poset cocalculus, a functor calculus framework for functors out of a poset, to study the problem of decomposing multipersistence modules into simpler components. We both prove new results in this topic and offer a new perspective…
The theory of zigzag persistence is a substantial extension of persistent homology, and its development has enabled the investigation of several unexplored avenues in the area of topological data analysis. In this paper, we discuss three…
This short note establishes explicit and broadly applicable relationships between persistence-based distances computed locally and globally. In particular, we show that the bottleneck distance between two zigzag persistence modules…
Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $1$-parameter persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…
We give down-to-earth proofs of the structure theorems for persistence modules.
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 -…
We characterize the class of persistence modules indexed over $\mathbb{R}^2$ that are decomposable into summands whose support have the shape of a {\em block}---i.e. a horizontal band, a vertical band, an upper-right quadrant, or a…