Geometric / Clifford algebra in arbitrary dimensions

[nileshtrivedi]

Geometric / Clifford algebra in arbitrary dimensions

[polvalente]

@nileshtrivedi I'm not aware of any libraries in that space (heh)
But you could build upon Nx for that. I read up a bit on Clifford Algebra and this is roughly where you would start for R2:

defmodule Geometric do
  defmodule Bivector do
    @moduledoc "represents a bivector with basis (e1, e2)"
    @derive {Nx.Container, containers: [:scalar]}
    defstruct [:scalar]
  end
  
  defmodule Multivector do
    @moduledoc "represents a multivector with a grade-0 (scalar) part and a grade-2 (bivector) part"
    @derive {Nx.Container, containers: [:scalar, :bivector]}
    defstruct [:scalar, :bivector]
  end
  
  import Nx.Defn
    
  defn wedge_product(%Nx.Tensor{shape: {n}} = v1, %Nx.Tensor{shape: {n}} = v2) do
    a = v1[0]
    b = v1[1]
    c = v2[0]
    d = v2[1]
    
    %Bivector{scalar: a * d - b * c}
  end
     
  defn geometric_product(%Nx.Tensor{shape: {n}} = v1, %Nx.Tensor{shape: {n}} = v2) do
    %MultiVector{scalar: Nx.dot(v1, v2), bivector: wedge_product(v1, v2)}
  end
end

Nx.Container allows you to define structs that work inside defn, and then you can compose things accordingly, even defining a custom add function that works differently based on the different input combinations.

[polvalente]

This is not currently possible in Nx, nor is planned.
For what it's worth, the example you showed only uses numpy for getting constants like e and pi, while the actual algebra is handled by the clifford library.

[nileshtrivedi]

@polvalente True. Are you aware of any elixir libraries that I can use in conjunction with Nx to do this kind of work?

[nileshtrivedi]

@polvalente Oh, thank you so much for taking the time to look into this (much more than I was expecting). This helps a lot. :-)