Euclidean distance and maximum likelihood retractions by homotopy continuation

We define a new second-order retraction map for statistical models. We also compute retractions using homotopy continuation. Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraint set, and hence retraction maps are used to approximate the exponential map, and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For other implicitly-defined manifolds or varieties, suitable retraction maps are difficult to compute. We therefore develop Algorithm 1, which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of $R^n$. We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. After defining an analogous maximum likelihood retraction, Algorithm 2 computes it using homotopy continuation. In Theorem 2, we prove that the resulting map is a second-order retraction; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.