Fast Global Optimization on the Torus, the Sphere, and the Rotation Group

Abstract

Detecting all local extrema or the global extremum of a polynomial on the torus, the sphere or the rotation group is a tough yet often requested numerical problem. We present a heuristic approach that applies common descent methods like nonlinear conjugated gradients or Newtons methods simultaneously to a large number of starting points. The corner stone of our approach are FFT like algorithms, i.e., algorithms that scale almost linearly with respect to the sum of the dimension of the polynomial space and the number of evaluation points. These FFT like algorithms allow us to compute one step of a descent method simultaneously for all staring points at almost the same cost as for one single starting point. The effectiveness of the proposed algorithms is demonstrated in various applications. In particular, we apply it to the Radon transform of a spherical function which allows us to detect lines in spherical patterns.

Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization

“…To simplify the presentation, in the sequel, we specialize to the case where X is a compact subset of R d with a periodic boundary condition. For instance, X can be a torus T d , which can be viewed as the d-dimensional hypercube [0, 1) d where the opposite (d − 1)-cells are identified, We specialize to such a structure only for rigorous theoretical analysis, which also appears in other works involving the Wasserstein space (Gräf and Hielscher, 2015). Our results can be readily generalized to a general X with extra technical care.…”