Related papers: Computing Generalized Ranks of 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 -…
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…
In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a $2$D persistence module…
In addition to inherent computational challenges, the absence of a canonical method for quantifying `persistence' in multi-parameter persistent homology remains a hurdle in its application. One of the best known quantifications of…
The theory of persistence modules on the commutative ladders $CL_n(\tau)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view…
A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if…
Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$…
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…
In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered…
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…
We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter…
One of the main objectives of topological data analysis is the study of discrete invariants for persistence modules, in particular when dealing with multiparameter persistence modules. In many cases, the invariants studied for these…
We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce…
In recent years, persistence modules have been viewed as graded modules with gradation over a preordered set serving as the indexing set. We provide sufficient criteria for a projective module over a PID to be free when the indexing set is…
In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic…
We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and…
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or,…
Given a graph whose edges are labeled by ideals in a ring, a generalized spline is a labeling of each vertex by a ring element so that adjacent vertices differ by an element of the ideal associated to the edge. We study splines over the…
Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which…