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Morse Theory for Filtrations and Efficient Computation of Persistent Homology
An efficient preprocessing algorithm is introduced to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups through an extension of combinatorial Morse theory from complexes to filtrations.
Simplicial Models and Topological Inference in Biological Systems
- Vidit Nanda, R. Sazdanovic
- Computer Science, MathematicsDiscrete and Topological Models in Molecular…
This article begins with the combinatorics and geometry of simplicial complexes and outline the standard techniques for imposing filtered simplicial structures on a general class of datasets, and computes topological statistics of the original data via the algebraic theory of (persistent) homology.
The Efficiency of a Homology Algorithm based on Discrete Morse Theory and Coreductions
Two implementations of a homology algorithm based on the Forman’s discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of…
Persistence Paths and Signature Features in Topological Data Analysis
- I. Chevyrev, Vidit Nanda, H. Oberhauser
- Computer ScienceIEEE Transactions on Pattern Analysis and Machine…
- 1 June 2018
A new feature map for barcodes as they arise in persistent homology computation that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.
Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps
- S. Harker, K. Mischaikow, M. Mrozek, Vidit Nanda
- Mathematics, Computer ScienceFound. Comput. Math.
- 1 February 2014
A new Morse theoretic preprocessing framework for deriving chain maps from set-valued maps is introduced, and hence an effective scheme for computing the morphism induced on homology by the approximated continuous function is provided.
Local Cohomology and Stratification
- Vidit Nanda
- MathematicsFound. Comput. Math.
- 2 July 2017
An algorithm to recover the canonical stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum.
The Space of Barcode Bases for Persistence Modules
A new algorithm for computing barcodes which also keeps track of, and outputs, such a change of basis and an explicit characterisation of the group of transformations that sends one barcode basis to another.
A topological measurement of protein compressibility
In this paper we partially clarify the relation between the compressibility of a protein and its molecular geometric structure. To identify and understand the relevant topological features within a…
Topological Measurement of Protein Compressibility via Persistence Diagrams
We exploit recent advances in computational topology to study the compressibility of various proteins found in the Protein Data Bank (PDB). Our fundamental tool is the persistence diagram, a…
Reconstructing Functions from Random Samples
From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high…