Related papers: Efficient Algorithms for Complexes of Persistence …
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…
This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to…
We present algorithms to compute the vector space of homomorphisms Hom(X,Y) between finitely generated representations of the partially ordered set Z^d. Our results generalise to any partially ordered set. Our main theoretical contribution…
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose…
We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use…
We introduce a refinement of persistent homology that detects simple-homotopy-theoretic phenomena invisible to homology. Given a filtered simplicial complex, we define the Morse complexity profile as the minimal number of critical simplices…
We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along…
Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. We…
In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
The topology of periodic spaces has attracted a lot of interest in recent years in order to study and classify crystalline structures and other large homogeneous data sets, such as the distribution of galaxies in cosmology. In practice,…
A multiplication on persistence diagrams is introduced by means of Schubert calculus. The key observation behind this multiplication comes from the fact that the representation space of persistence modules has the structure of the Schubert…
Sheaf cohomology or, more generally, higher direct images of coherent sheaves along proper morphisms are central to modern algebraic geometry. However, the computation of these objects is a non-trivial and expensive task which easily…
We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…
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…
Persistent homology (PH) is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general…
We use Topological Data Analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify lifetime of…
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input…
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…