manifolds.jl

abstract type Manifold
end

# Retraction type should be a part of the type of manifold,
# either by different concrete types or a single type with a type parameter
# that selects the retraction type.
retr(M::Manifold, v, x, t=1) = error("Not implemented")
retr!(M::Manifold, y, v, x, t=1) = error("Not implemented")

# just a reasonable default
distance(M::Manifold, x, y) = norm(log(M, x, y))

# inner product of ξ and ν that are tangent vectors at x
# Manopt.jl and ManifoldProjections.jl calls this dot, Manopt and OptimKit use inner, FunManifolds uses innerproduct
dot(M::Manifold, x, ξ, ν) = error("Not implemented")
norm(M::Manifold, x, v) = sqrt(inner(M, x, v, v))

# Proper exponential map
# mutating version saves the result to y
exp(M::Manifold, x, ξ, t=1) = error("Not implemented")
exp!(M::Manifold, y, x, ξ, t=1) = error("Not implemented")

# Proper logarithmic map
log(M::Manifold, x, y) = error("Not implemented")
log!(M::Manifold, v, x, y) = error("Not implemented")

# dimension of given manifold
manifold_dimension(M::Manifold) = error("Not implemented")

# type of vector transport (isometric, some approximation or something else)
# should be a part of the type of manifold,
# either by different concrete types or a single type with a type parameter
# that selects the retraction type.
vector_transport(M::Manifold, xfrom, xat, v) = error("Not implemented")
vector_transport!(M::Manifold, vto, xfrom, xat, v) = error("Not implemented")

random_point(M::Manifold, options...) = error("Not implemented")
random_tvector(M::Manifold, x, options...) = error("Not implemented")

typical_distance(M::Manifold) = error("Not implemented")
zero_tvector(M::Manifold, x) = log(M, x, x)
zero_tvector!(M::Manifold, v, x) = log!(M, v, x, x)

geodesic(M::Manifold, x, y, t) = exp(M, x, log(M, x, y), t)

Then its underscores for functions.

For Manopt.jl my idea was to do both (but I think I wrote that) project(M,N,x) and embed(N,M,y) but only really of both are known. Whether that can be included in our common interface, I am not sure.

Wy remove retract? I favour or retr? I'd be fine with that.

For Manopt.jl my idea was to do both (but I think I wrote that) project(M,N,x) and embed(N,M,y) but only really of both are known. Whether that can be included in our common interface, I am not sure.

I think the safe choice would be to wait with project and embed until there is a clear use case. We can always add new functions.

Wy remove retract? I favour or retr? I'd be fine with that.

Yes, I think retr should be sufficient.

retr is sufficient.

Yes, I'll work on that as one of my next features, so I'll provide an update, when I have more details on that and for now we should omit those two.