  We reduce a broad class of machine learning problems, usually addressed by EM
or sampling, to the problem of finding the $k$ extremal rays spanning the
conical hull of a data point set. These $k$ "anchors" lead to a global solution
and a more interpretable model that can even outperform EM and sampling on
generalization error. To find the $k$ anchors, we propose a novel
divide-and-conquer learning scheme "DCA" that distributes the problem to
$\mathcal O(k\log k)$ same-type sub-problems on different low-D random
hyperplanes, each can be solved by any solver. For the 2D sub-problem, we
present a non-iterative solver that only needs to compute an array of cosine
values and its max/min entries. DCA also provides a faster subroutine for other
methods to check whether a point is covered in a conical hull, which improves
algorithm design in multiple dimensions and brings significant speedup to
learning. We apply our method to GMM, HMM, LDA, NMF and subspace clustering,
then show its competitive performance and scalability over other methods on
rich datasets.
