Computational Homology

This package provides various computational homology tools for cellular complexes.

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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