Let M in R^d denote a low-dimensional manifold and let X={x1, …, xn} be a collection of points uniformly sampled from M. We study the relationship between the curvature of a random geometric graph built from M and the curvature of the manifold M via continuum limits of Ollivier’s discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if M has Ricci curvature bounded from below by a positive constant, then the random geometric graph will inherit this global structural property with high probability. We discuss applications of the global discrete curvature bounds to contraction properties of heat kernels on graphs, as well as implications for manifold learning from data clouds. In particular, we show that the consistency results allow for characterizing the intrinsic curvature of a manifold from extrinsic curvature.

We study geodesically convex (g-convex) problems that can be written as a difference of Euclidean convex functions. This structure arises in key applications such as matrix scaling, M-estimators of scatter matrices, and Brascamp-Lieb inequalities. In particular, we exploit this structure to make use of the Convex-Concave Procedure (CCCP), which helps us bypass potentially expensive Riemannian operations and leads to very competitive solvers. Importantly, unlike existing theory for CCCP that ensures convergence to stationary points, we exploit the overall g-convexity structure and provide iteration complexity results for mph{global optimality}. We illustrate our results by specializing them to a few concrete optimization problems that have been previously studied in the machine learning literature. We hope our work spurs the study of mixed Euclidean-Riemannian optimization algorithms.

We study projection-free methods for constrained Riemannian optimization. In particular, we propose a Riemannian Frank-Wolfe (RFW) method that handles constraints directly, in contrast to prior methods that rely on (potentially costly) projections. We analyze non-asymptotic convergence rates of RFW to an optimum for geodesically convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize RFW to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian “linear” oracle required by RFW admits a closed form solution; this result may be of independent interest. We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods, and observe that RFW performs competitively on the task of computing Riemannian centroids.

We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a cost function posed over a finite time interval, in contrast to much previous work that considers asymptotics as the time horizon tends to infinity. We study several different versions of the problem, including Bayesian control, in which we assume a prior distribution on the unknown parameter; and “agnostic” control, in which we assume nothing about the unknown parameter. For the agnostic problems, we compare our performance with that of an opponent who knows the value of the parameter. This comparison gives rise to several notions of “regret,” and we obtain strategies that minimize the “worst-case regret” arising from the most unfavorable choice of the unknown parameter. In every case, the optimal strategy turns out to be a Bayesian strategy or a limit of Bayesian strategies.

We introduce Forman-Ricci curvature and its corresponding flow as characteristics for complex networks attempting to extend the common approach of node-based network analysis by edge-based characteristics. Following a theoretical introduction and mathematical motivation, we apply the proposed network-analytic methods to static and dynamic complex networks and compare the results with established node-based characteristics. Our work suggests a number of applications for data mining, including denoising and clustering of experimental data, as well as extrapolation of network evolution.