Many machine learning applications involve non-Euclidean data, such as graphs, strings or matrices. In such cases, exploiting Riemannian geometry can deliver algorithms that are computationally superior to standard(Euclidean) nonlinear programming approaches. This observation has resulted in an increasing interest in Riemannian methods in the optimization and machine learning community.

In the first part of the talk, we consider the task of learning a robust classifier in hyperbolic space. Such spaces have received a surge of interest for representing large-scale, hierarchical data, due to the fact that theyachieve better representation accuracy with fewer dimensions. We present the first theoretical guarantees for the (robust) large margin learning problem in hyperbolic space and discuss conditions under which hyperbolic methods are guaranteed to surpass the performance of their Euclidean counterparts. In the second part, we introduce Riemannian Frank-Wolfe (RFW) methods for constrained optimization on manifolds. Here, we discuss matrix-valued tasks for which such Riemannian methods are more efficient than classical Euclidean approaches. In particular, we consider applications of RFW to the computation of Riemannian centroids and Wasserstein barycenters, both of which are crucial subroutines in many machine learning methods.

From latent variable models in machine learning to inverse problems in computational imaging, tensors pervade the data sciences.  Often, the goal is to decompose a tensor into a particular low-rank representation, thereby recovering quantities of interest about the application at hand.  In this talk, I will present a recent method for low-rank CP symmetric tensor decomposition.  The key ingredients are Sylvester’s catalecticant method from classical algebraic geometry and the power method from numerical multilinear algebra.  In simulations, the method is roughly one order of magnitude faster than existing CP decomposition algorithms, with similar accuracy.  I will state guarantees for the relevant non-convex optimization problem, and robustness results when the tensor is only approximately low-rank (assuming an appropriate random model).  Finally, if the tensor being decomposed is a higher-order moment of data points (as in multivariate statistics), our method may be performed without explicitly forming the moment tensor, opening the door to high-dimensional decompositions.  This talk is based on joint works with João Pereira, Timo Klock and Tammy Kolda. 

One of the most successful methods in topological data analysis (TDA) is persistent homology, which associates a one-parameter family of spaces to a data set, and gives a summary — an invariant called "barcode" — of how topological features, such as the number of components, holes, or voids evolve across the parameter space. In many applications one might wish to associate a multiparameter family of spaces to a data set. There is no generalisation of the barcode to the multiparameter case, and finding algebraic invariants that are suitable for applications is one of the biggest challenges in TDA.

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the study of distances for multiparameter persistence modules is more challenging, as they rely on a choice of suitable invariant.

In this talk I will first give a brief introduction to multiparameter persistent homology. I will then present a general framework to study stability questions in multiparameter persistence: I will discuss which properties we would like invariants to satisfy, present different ways to associate distances to such invariants, and finally illustrate how this framework can be used to derive new stability results. No prior knowledge on the subject is assumed.

The talk is based on joint work with Barbara Giunti, John Nolan and Lukas Waas. 

A topology optimization problem for Stokes flow finds the optimal material distribution of a Stokes fluid that minimizes the fluid’s power dissipation under a volume constraint. In 2003, T. Borrvall and J. Petersson [1] formulated a nonconvex optimization problem for this objective. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this talk, we will extend and refine their numerical analysis. We will show that there exist finite element functions, satisfying the necessary first-order conditions of optimality, that converge strongly to each isolated local minimizer of the problem.

[1] T. Borrvall, J. Petersson, Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77–107. doi:10.1002/fld.426.

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