The functions spd.logmap() and spd.expmap() implement the logarithm and exponential maps, respectively, at a basepoint p. Given spd.matrices x and p, spd.logmap(x, p) returns a tangent vector (a symmetric matrix) located at p, equal to the projection of x onto the tangent space at p. Often, this operation is interpreted as the linearization of the SPD manifold around p. The projection of a tangent vector y at p back onto the SPD manifold is accomplished using spd.expmap(y, p).

Transport

spdm can transport tangent vectors between tangent spaces using the function spd.transport(x, from, to, method), where x is a symmetric matrix (a tangent vector), and from and to are SPD matrices. The following two forms of transport are currently implemented:

Parallel Transport

Setting method = "pt" uses the Schild’s ladder algorithm to parallel transport a tangent vector along a geodesic between two SPD matrices. An additional argument nsteps sets the number of steps used in the Schild’s ladder algorithm.

spd.transport(x, from, to, method = 'pt', nsteps = 10)

GL(n) Action

The metric on the Riemannian manifold \(\mathcal{S}_{++}\) of SPD matrices is invariant under the \(GL_n(\mathbb{R})\) action

\[\phi: GL_n(\mathbb{R}) \times \mathcal{S}_{++} \rightarrow \mathcal{S}_{++}\] \[\phi_G(\Sigma) = G \Sigma G^\top\]

Zhao et al. (2018) propose to transport a tangent vector \(V\) from \(S_1\) to \(S_2\) using the differential

\[d\phi_G(V) = G V G^\top,\]

and setting \(G = S_2^{1/2}S_1^{-1/2}\), giving the transport

\[d\phi_{S_1 \rightarrow S_2}(V) = S_2^{1/2} S_1^{-1/2} V S_1^{-1/2} S_2^{1/2}\]

This can be done by setting method = "gl"

spd.transport(x, from, to, method = 'gl')

References

Zhao, Q., Kwon, D., & Pohl, K. M. (2018, September). A Riemannian Framework for Longitudinal Analysis of Resting-State Functional Connectivity. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 145-153). Springer, Cham.