Related papers: Parallel computation of interval bases for persist…
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…
Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that…
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,…
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…
Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…
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…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
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…
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…
We give down-to-earth proofs of the structure theorems for persistence modules.
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Time-series of persistence diagrams, known as vineyards, have shown to be useful in diverse applications. A natural algebraic version of vineyards is a time series of persistence modules equipped with interleaving maps between the…
Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many…
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…
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…
The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special…
While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and…
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…
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…
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…