## Riemannian operations

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.