pdParTrans: Riemannian HPD parallel transport

In JorisChau/pdSpecEst: An Analysis Toolbox for Hermitian Positive Definite Matrices

Description

pdParTrans computes the parallel transport on the manifold of HPD matrices equipped with the affine-invariant Riemannian metric as described in e.g., Chapter 2 of \insertCiteC18pdSpecEst. That is, the function computes the parallel transport of a Hermitian matrix W in the tangent space at the HPD matrix P along a geodesic curve in the direction of the Hermitian matrix V in the tangent space at P for a unit time step.

Usage

pdParTrans(P, V, W)

Arguments

P	a (d,d)-dimensional HPD matrix.
V	a (d,d)-dimensional Hermitian matrix corresponding to a vector in the tangent space of P.
W	a (d,d)-dimensional Hermitian matrix corresponding to a vector in the tangent space of P.

Value

a (d,d)-dimensional Hermitian matrix corresponding to the parallel transportation of W in the direction of V along a geodesic curve for a unit time step.

References

\insertAllCited

See Also

Expm, Logm

Examples

## Transport the vector W to the tangent space at the identity
W <- matrix(complex(real = rnorm(9), imaginary = rnorm(9)), nrow = 3)
diag(W) <- rnorm(3)
W[lower.tri(W)] <- t(Conj(W))[lower.tri(W)]
p <- matrix(complex(real = rnorm(9), imaginary = rnorm(9)), nrow = 3)
P <- t(Conj(p)) %*% p

pdParTrans(P, Logm(P, diag(3)), W) ## whitening transport

JorisChau/pdSpecEst documentation built on Jan. 13, 2020, 6:08 p.m.