PersistentCohomology

Pure Julia package to compute persistent cohomology. In particular it implements the persistent cocycle algorithm from Persistent Cohomology and Circular Coordinates - Vin de Silva, Dmitriy Morozov, Mikael Vejdemo-Johansson. The algorithm can be used with coefficients on an arbitrary field: finite fields are available in Julia via the GaloisFields package.

This package provides a type to hold cochains (called Cochain) and two functions: vietorisrips to compute the filtered Vietoris Rips complex from a sparse distance matrix and persistentcocycles to compute persistent cohomolofy on such complex. See the docstrings of Cochain, vietorisrips and persistentcocycles for more details.

Install

To install this package (and GaloisFields which you'll probably need as well):

pkg> add https://github.com/piever/PersistentCohomology.jl.git

pkg> add GaloisFields

Examples

julia> using PersistentCohomology, SparseArrays, GaloisFields

julia> M = sparse(diagm(1 => fill(sqrt(2), 3), 2 => fill(2.0, 2), 3 => [sqrt(2)]))
4×4 SparseMatrixCSC{Float64,Int64} with 6 stored entries:
  [1, 2]  =  1.41421
  [1, 3]  =  2.0
  [2, 3]  =  1.41421
  [1, 4]  =  1.41421
  [2, 4]  =  2.0
  [3, 4]  =  1.41421

julia> cplx = vietorisrips(M, 2)
(Float64-valued 0-Cochain, Float64-valued 1-Cochain, Float64-valued 2-Cochain)

julia> const F3 = @GaloisFields 3

julia> cocycles, spans = persistentcocycles(F3, cplx, 1);

julia> cocycles[1]
4-element Array{Cochain{StructArrays.StructArray{Tuple{Int64},1,NamedTuple{(:x1,),Tuple{Array{Int64,1}}}},SparseVector{𝔽₃,Int64}},1}:
 Sparse 𝔽₃-valued 0-Cochain. Non-zero values:
(1,) => 1
(2,) => 1
(3,) => 1
(4,) => 1

 Sparse 𝔽₃-valued 0-Cochain. Non-zero values:
(2,) => 1

 Sparse 𝔽₃-valued 0-Cochain. Non-zero values:
(3,) => 1

 Sparse 𝔽₃-valued 0-Cochain. Non-zero values:
(4,) => 1


julia> spans[1]
4-element Array{IntervalSets.Interval{:closed,:open,Float64},1}:
 0.0..Inf (closed–open)
 0.0..1.4142135623730951 (closed–open)
 0.0..1.4142135623730951 (closed–open)
 0.0..1.4142135623730951 (closed–open)

julia> cocycles[2]
1-element Array{Cochain{StructArrays.StructArray{Tuple{Int64,Int64},1,NamedTuple{(:x1, :x2),Tuple{Array{Int64,1},Array{Int64,1}}}},SparseVector{𝔽₃,Int64}},1}:
 Sparse 𝔽₃-valued 1-Cochain. Non-zero values:
(4, 3) => 1


julia> spans[2]
1-element Array{IntervalSets.Interval{:closed,:open,Float64},1}:
 1.4142135623730951..2.0 (closed–open)