Quaternions and Euler angles are discontinuous and difficult for neural networks to learn. They show 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. i.e. regress two vectors and apply Graham-Schmidt (GS).

@maurice_weiler @lcfalors @pimdehaan @im_td very interesting, thanks for highlighting this to me!

@maurice_weiler @ankurhandos @lcfalors @pimdehaan @im_td The map I: : S^2 \times S^2 --> SO(3) is not continuous: it has a singularity (and is not properly.defined) on the diagonal {(v, v)} and the anti diagonal {(v, -v)}.

@RogierBrussee @ankurhandos @lcfalors @pimdehaan @im_td This is true. But the restriction of pi to the image of i should be fine?

@maurice_weiler @ankurhandos @lcfalors @pimdehaan @im_td could these ideas be extended to dimension higher than three? It seems that the construction of the orthornomal basis {w_1, w_2,w_3} is very specific to d=3. What am I missing?

@SaraASolla @ankurhandos @lcfalors @pimdehaan @im_td yes, the same idea is in principle easily applicable to SO(N): You predict N-1 (linearly independent) vectors in R^N and orthonormalize them. Then you add the unique N-th vector such that you get a right handed orthonormal frame which corresponds to a group element of SO(N).

@maurice_weiler @ankurhandos @lcfalors @pimdehaan @im_td Wow cool read! Homeomorphisms and VAEs? Sign me up