TSC

Tangent space can be used to manipulate data (filtering or classification). Data are mapped to the tangent space using Logarithmic map

\forall i, S_i= \log_P(P_i),

with P the Geometric mean of all data.
Since Tangent space are Euclidean, it's possible to treat the S_i as vectors (taking care of removing symmetric elements)

S_i=vect(S_i).

it's possible to apply classical methods on the \tilde{S_i} and if necessary, go back to the Riemannian Space.

P_i= \exp_P(unvect(\tilde{S_i}))

Fig.6: graphical example of TSC

It is used any classifier such as LDA on the vectorized upper-triangular elements of the projected matrices.

Bibliography

A. Barachant, S. Bonnet, M. Congedo and C. Jutten, "Multiclass brain–computer interface classification by Riemannian geometry", IEEE Trans. Biomed. Eng., vol. 59, no. 4, pp. 920-928, 2012.