Selected Publications

More Publications

We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic Riemannian Frank-Wolfe methods for nonconvex and geodesically convex problems. We present algorithms for both purely stochastic optimization and finite-sum problems. For the latter, we develop variance-reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two classic tasks: The computation of the Karcher mean of positive definite matrices and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.
IMA Journal of Numerical Analysis, 2021.

We exhibit optimal control strategies for settings 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. In the agnostic case, we compare our performance with that of an opponent who knows the value of the parameter. In every case, the optimal strategy turns out to be a Bayesian strategy or a limit of Bayesian strategies.
Rev. Mat. Iberoamericana, 2021.

We present, to our knowledge, the first theoretical guarantees for learning a classifier in hyperbolic rather than Euclidean space. Specifically, we consider the problem of learning a large-margin classifier for data possessing a hierarchical structure. Our first contribution is a hyperbolic perceptron algorithm, which provably converges to a separating hyperplane. We then provide an algorithm to efficiently learn a large-margin hyperplane, relying on the careful injection of adversarial examples. Finally, we prove that for hierarchical data that embeds well into hyperbolic space, the low embedding dimension ensures superior guarantees when learning the classifier directly in hyperbolic space.
NeurIPS, 2020.

The problem of identifying geometric structure in heterogeneous, high-dimensional data is a cornerstone of representation learning. We propose a combinatorial approach to evaluating embeddability, i.e., to decide whether a data set is best represented in Euclidean, Hyperbolic or Spherical space. Our method analyzes nearest-neighbor structures and local neighborhood growth rates to identify the geometric priors of suitable embedding spaces. For canonical graphs, the algorithm’s prediction provably matches classical results. As for large, heterogeneous graphs, we introduce an efficiently computable statistic that approximates the algorithm’s decision rule.
AISTATS, 2020.

Recent and Upcoming Talks

More Talks

Riemannian Frank-Wolfe Methods

May 20, 2021, SIAM Conference on Applied Linear Algebra (Minisymposium: Linear Algebra and Differential Geometry)

Geometric Methods for Machine Learning and Optimization

Jan 20, 2021, One World Seminar Series: Mathematics of Machine Learning

Riemannian Frank-Wolfe Methods

Jan 5, 2021, Joint Mathematics Meeting (AMS Special Session: Geometry in the Mathematics of Data Science)

Learning a robust classifier in hyperbolic space

Dec 11, 2020, NeurIPS Workshop Differential Geometry meets Deep Learning

Selected Projects

Learning & Control Theory

Learning in the sparse data regime.

Riemannian Frank-Wolfe Methods

Nonconvex and g-convex optimization on manifolds.

Curvature-based Network Analysis

Forman-Ricci curvature and discrete geometric flows for a curvature-based analysis of complex networks.

Machine Learning in Non-Euclidean Spaces

Understanding and Learning the Geometry of Data.