Part 11 of 15. A Voronoi diagram cuts a space into cells by nearest-neighbour membership. On a statistical manifold, the cells warp into shapes that a Euclidean eye would call impossible, and the shapes are exactly right for clustering distributions.
Play: Drag any of the sites to see how different definitions of "distance" partition the Gaussian manifold. Notice how the Euclidean metric produces flat polygons, while KL and Fisher-Rao metrics produce curved, warped cells that respect the underlying hyperbolic geometry of probability distributions.
A Voronoi diagram partitions a space by assigning every point to its "nearest" site. But on a statistical manifold, what does "nearest" mean? If you use Euclidean distance, you get straight lines and convex polygons, but this fundamentally misunderstands the geometry of probability. A unit step in at is statistically vastly different from one at .
If we instead define distance using a Bregman divergence (like KL) or a Riemannian geodesic distance (like Fisher-Rao), the Voronoi cells warp to reflect the true geometry of the space:
Because KL is a Bregman divergence, the KL Voronoi diagram on a curved manifold is mathematically equivalent to a flat power diagram in expectation coordinates. This means we can compute these seemingly complex curved partitions just as fast as standard Euclidean ones.
Part 12 reads the EM algorithm as alternating projection between two flat submanifolds — Amari, 1980s. The Pythagorean theorem gives EM's monotone convergence in one line.