Related papers: Discrete Morse Theory for Computing Zigzag Persist…
Using Banchoff's discrete Morse Theory, in tandem with Bloch's result on the strong connection between the former and Forman's Morse Theory, and our own previous algorithm based on the later, we show that there exists a curvature-based,…
Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has…
Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices…
For a $P$-indexed persistence module ${\sf M}$, the (generalized) rank of ${\sf M}$ is defined as the rank of the limit-to-colimit map for the diagram of vector spaces of ${\sf M}$ over the poset $P$. For $2$-parameter persistence modules,…
Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to…
This is a survey article on the recent progress in understanding the Strominger-Yau-Zaslow (SYZ) mirror symmetry conjecture, especially on the effect of quantum corrections, via Witten-Morse theory using the program first depicted by Fukaya…
In this note one shows that the four persistence diagrams and measures defined in [5] are particular cases of the maps and the measures discussed in [2] and [1]
We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type $\widetilde{\mathbb{A}}$ quivers, there is a precise relationship between the associated…
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 solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse…
Persistent homology and zigzag persistent homology are techniques which track the homology over a sequence of spaces, outputting a set of intervals corresponding to birth and death times of homological features in the sequence. This paper…
Persistent homology tracks topological features across geometric scales, encoding birth and death of cycles as barcodes. We develop a complementary theory where the filtration parameter is algebraic precision rather than geometric scale.…
It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently…
We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$,…
This paper introduces a reformulation of the classical convergence theorem for spectral sequences of filtered complexes which provides an algorithm to effectively compute the induced filtration on the total (co)homology, as soon as the…
Reservoir computing is a well-established approach for processing data with a much lower complexity compared to traditional neural networks. Despite two decades of experimental progress, the core properties of reservoir computing (namely…
A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning…
Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to…
In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare…
The main objective of this paper is to extend Morse-Forman theory to vector-valued functions. This is mostly motivated by the need to develop new tools and methods to compute multiparameter persistence. To generalize the theory, in addition…