Basic usage

You can calculate Riemannian gradient of a function defined in its embedding in multiple ways. DifferentiationInterface.jl can be used to select the backend.

using ManifoldDiff
using DifferentiationInterface
using Manifolds, FiniteDifferences, ForwardDiff, Zygote

rb_onb_fd51 = TangentDiffBackend(AutoFiniteDifferences(central_fdm(5, 1)))
rb_onb_fwdd = TangentDiffBackend(AutoForwardDiff())
rb_proj_zyg = RiemannianProjectionBackend(AutoZygote())

s2 = Sphere(2)

A = [1.0 2.0 5.0; 2.0 -1.0 4.0; 5.0 4.0 0.0]

f(p) = p' * A * p
q = [0.0, 1.0, 0.0]

println(ManifoldDiff.gradient(s2, f, q, rb_onb_fd51))
println(ManifoldDiff.gradient(s2, f, q, rb_onb_fwdd))
println(ManifoldDiff.gradient(s2, f, q, rb_proj_zyg))

[3.999999999999511, 0.0, 7.999999999999624]
[4.0, 0.0, 8.0]
[4.0, 0.0, 8.0]

In this example rb_onb_fd51 corresponds to a finite differencing scheme, rb_onb_fwdd calculates gradient using ForwardDiff.jl and rb_proj_zyg uses Zygote.jl for reverse mode automatic differentiation.

TangentDiffBackend reduces dimensionality of the problem to the intrinsic dimension of the manifold, while RiemannianProjectionBackend relies on converting Euclidean gradient in the embedding to the Riemannian one.