Transforms for complex manifolds

[sethaxen]

A few of the manifolds we have (the sphere, symmetric positive definite matrices, and Stiefel manifold) have straightforward complex analogs that appear in statistical applications, and the transforms for these manifolds can be generalized to map to these complex spaces with likely very little additional effort. I suggest we do so for completeness since Stan now has complex number support.

Since I've been working on these manifolds, I would take care of this generalization, but before putting in the effort, I want to confirm if others think this could be within the scope of the paper.

A few notes:

• the complex sphere ℂⁿ is equivalent to the real sphere ℝ²ⁿ, so we don't need to do anything special here.
• symmetric positive definite matrices generalize to hermitian positive definite matrices. Everything is the same, except eigenvectors are now unitary instead of orthogonal, and the sub-diagonal of the lower Cholesky factor is complex. We could test with the complex Wishart distribution
• Stiefel: the semi-orthogonal matrices become semi-unitary. This allows for potentially better transforms. e.g. the Orthogonal group O(N) is a subgroup of unitary U(N), but it has 2 disconnected submanifolds (the ones with +1 and -1 determinant). Thus any transform from unconstrained space to O(N) must use parameter expansion to connect the 2 submanifolds; bijective maps like matrix exponential can only cover one of the submanifolds. However, unitary matrices have only the constraint that the absolute value of the determinant is 1 (i.e. it the determinant lives on the circle in the complex plane), which smoothly connects the submanifolds with +1 and -1 determinant, so parameter expansion is not needed, and bijective maps are possible. Distributions like Matrix von Mises-Fisher and Bingham have complex analogs we could test with.

[bob-carpenter]

That sounds super cool. I think we'll have enough in this paper without the complex cases, but I'm amenable to adding them with one caveat: that we do it after we have an end to end draft of the paper that only discusses the simplex case. I want to treat this paper like agile software development and build up from a minimally working example that establishes notation and evaluation scheme. Otherwise, I fear adding more math in everyone's individual notations will make it impossible to finish the paper.

[sethaxen]

I'm amenable to adding them with one caveat: that we do it after we have an end to end draft of the paper that only discusses the simplex case.

Okay!

I want to treat this paper like agile software development and build up from a minimally working example that establishes notation and evaluation scheme. Otherwise, I fear adding more math in everyone's individual notations will make it impossible to finish the paper.

Oops! I've been collecting a lot of transforms and determinants. I can pause for now.

[spinkney]

This paper https://arxiv.org/abs/1812.07685 has the construction random correlation matrices for complex and quaternion entries. As well as the hyperspherical stuff