/ComputationalHomology.jl

This package provides various computational homology tools for cellular complexes.

Primary LanguageJuliaOtherNOASSERTION

Computational Homology

This package provides various computational homology tools for cellular complexes.

Documentation Build Status

Installation

For Julia 1.1+, add BoffinStuff registry in package manager, and proceed with installation:

pkg> registry add https://github.com/wildart/BoffinStuff.git
pkg> add ComputationalHomology

Features

  • Cells

    • Simplex
    • Cube
    • CW

  • Chains for specified PID

  • Complexes

    • Simplicial
    • CW

  • Filtrations

  • Constructions

    • Čech
    • Vietoris-Rips
    • Witness

  • Homology

    • Simplicial
    • Persistent

  • Persistence

    • Barcodes / Diagrams
    • Persistence Landscape
    • Persistence Image
    • Distances
      • Wasserstein
    • Tropical Coordinates

Example

julia> using ComputationalHomology

julia> X = rand(3,10); # generate dataset

julia> cplx, w = vietorisrips(X, 0.4) # generate Vietoris-Rips (VR) complex
(SimplicialComplex((10, 12, 4)), Dict(0=>[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0],2=>[0.338893, 0.28014, 0.384243, 0.380966],1=>[0.338893, 0.321811, 0.304665, 0.310862, 0.27196, 0.28014, 0.366947, 0.380966, 0.191768, 0.384243, 0.359153, 0.365016]))

julia> flt = filtration(cplx, w) # construct filtration complex from VR complex
Filtration(SimplicialComplex((10, 12, 4)))

julia> ph = persistenthomology(flt, TwistReduction) # create persistent homology object with specific computation method
PersistentHomology[Filtration(SimplicialComplex((10, 12, 4))) with ComputationalHomology.TwistReduction]

julia> group(ph, 0) # calculate 0-homology group
2

julia> group(ph, 1) # calculate 1-homology group
3

TODO

  • Distances for persistence diagrams
  • Landscape standard deviation