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Topcat - a library for multiparameter persistence
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README.md

TOPCAT

Topcat is Java library for computing and analyzing multiparameter persistence modules. It comes with a python interface and a jupyter notebook with examples of how to use it.

Input to Topcat can be given as either:

  • a list of distance matrices and a list of filtration values for each parameter
  • a multifiltered simplicial complex

The current implementation of Topcat has been inspired by Ulrich Bauers work on Ripser (http://ripser.org), where he uses the following optimizations (among others):

  • computes the cohomology instead of homology,
  • uses the combinatorial number system to index simplices,
  • performs an implicit matrix reduction of the coboundary matrix .

We make heavy use of these techniques in Topcat to be able to handle millions of simplices.


A detailed description of the algorithms implemented in Topcat to compute the multiparameter persistence modules can be found in my master's thesis. A description of the theory of persistence contours, noise systems and the stable rank can be found in the following papers:

  • "Multidimensional Persistence and Noise" by Scolamiero et al. (arXiv:1505.06929)
  • "Stable Invariants for Multidimensional Persistence" by G and Chachólski (arXiv:1703.03632)

Future work:

  • Make use of the Chunk reduction described in the paper ''Chunk Reduction for Multi-Parameter Persistent Homology'' by Fugacci and Kerber (arXiv:1812.08580),
  • Implement multiparameter persistence landscapes as described in the paper ''Multiparameter Persistence Landscapes'' by Oliver Vipond (arXiv:1812.09935).

Requirements

  • Maven 2/3
  • Java 7 or higher
  • Python 2.7/3 (optional)
  • py4j (optional)

Installation

In the project root folder, compile the java code with Maven using the following:

mvn clean install

Usage

The library can be accessed either within java or via the Python-interface.

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