Algebraic analysis of rotation data

Nonlinear algebra and applications

“…for some λ i ∈ R. This implies that smooth optimal solutions to (3) are to be found among the solutions of the system of polynomial equations (4). From now on we move to the field of complex numbers and hence consider also complex solutions to (4).…”

“…for some λ i ∈ R. This implies that smooth optimal solutions to (3) are to be found among the solutions of the system of polynomial equations (4). From now on we move to the field of complex numbers and hence consider also complex solutions to (4). As it has been discussed in Section 0, this is a key technical condition to employ algebraic methods, as it allows us to study the generic instance of our problem.…”

“…For some functions, those methods are encoded in the software package hgm [234] in R [198]. The said methods were applied for instance to the inference of rotation data sampled according to the Fisher distribution [212,139,4] and to data arising from medical imaging.…”

“…In Figure 4, right, we plot its orthogonal projection onto a random three-dimensional subspace of its ambient R 4 . Each point p ∈ C ⊆ R 4 corresponds bijectively to a distinct placement p : V → R 2 , which is a map placing the vertices of its underlying graph (V, E) with V = {1, 2, 3, 4} and E = {(1, 2), (1, 4), (2, 3), (3,4)} in the plane R 2 , while preserving the lengths of its edges. We state the general setup below.…”

Nonlinear Algebra and Applications

“…for some λ i ∈ R. This implies that smooth optimal solutions to (3) are to be found among the solutions of the system of polynomial equations (4). From now on we move to the field of complex numbers and hence consider also complex solutions to (4). As it has been discussed in Section 0, this is a key technical condition to employ algebraic methods, as it allows to study the generic instance of our problem.…”