Related papers: Probabilistic Analysis of Multiparameter Persisten…
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…
A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is…
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 exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would permit multi-parameter…
It has been shown that $1$-parameter persistence modules have a very simple classification, namely there is a discrete invariant called a barcode that completely characterizes $1$-parameter persistence modules up to isomorphism. In…
The output of persistent homology is an algebraic object called a persistence module. This object admits a decomposition into a direct sum of interval persistence modules described entirely by the barcode invariant. In this paper we…
In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the…
A recent work by Lesnick and Wright proposed a visualisation of $2$D persistence modules by using their restrictions onto lines, giving a family of $1$D persistence modules. We give a constructive proof that any $1$D persistence module with…
In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called…
We consider the problem of multivariate density deconvolution when the interest lies in estimating the distribution of a vector-valued random variable but precise measurements of the variable of interest are not available, observations…
Multiparameter persistence module can capture more topological differences across data instances compared to using a single parameter, where the well-studied matching distance investigates the distance along a straight line in 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,…
We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are $1$-interleaved is NP-complete, already for bigraded, interval…
We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of…
We identify the measurable absorbing obstruction to uniqueness of invariant probability measures for a Markov kernel. Ordinary absorbing decompositions obstruct global irreducibility and recurrence, but not necessarily uniqueness: an…
We give a decomposition as a direct sum of indecomposable modules of several types of Specht modules in characteristic $2$. These include the Specht modules labelled by hooks, whose decomposability was considered by Murphy. Since the main…
Graphcodes were recently introduced as a technique to employ two-parameter persistence modules in machine learning tasks (Kerber and Russold, NeurIPS 2024). We show in this work that a compressed version of graphcodes yields a description…
Using a result by Chazal, Crawley-Boevey and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper…
The bedrock of persistence theory over a single parameter is decomposition of persistence modules into intervals. In [HLM24], the authors leveraged interval decomposition to produce a cell decomposition of the minimal model of a simply…