[MITSardine]

Cool article. Regarding section "The exponential map and logarithm map", if you're interested in computing the matrix exponential, there is the classic: C Moler, C Van Loan, "Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later". Also, using the series expansion is not necessarily unrobust as long as you don't stop at a fixed number of iterations but instead go on as long as terms have a norm greater than some tolerance. Scaling and squaring can be used to remain always in a given range of norms (less than 1, say).

Regarding Pitfall #3, the interpolation scheme exp(tlog(A) + (1-t)log(B)) is shortest path in a sense, just not with the usual matrix norms. See V Arsigny et al., "Log‐Euclidean metrics for fast and simple calculus on diffusion tensors". I can't help but find it more elegant than exp(log(BA^{-1})t)A which could just as well have been exp(log(A^{-1}B)t)A, or even Aexp(log(A^{-1}B)t), right? It also fixes the "no more than two transforms", as you can put any convex combination in exp(sum_i x_i log(A_i)).