Selected Publications

We study projection free methods for constrained geodesically convex optimization. In particular, we propose a Riemannian version of the Frank-Wolfe (RFW) method. We analyze RFW’s convergence and provide a global, non-asymptotic sublinear convergence rate. We also present a setting under which RFW can attain a linear rate. Later, we specialize RFW to the manifold of positive definite matrices, where we are motivated by the specific task of computing the geometric mean (also known as Karcher mean or Riemannian centroid). For this task, RFW requires access to a “linear oracle” that turns out to be a nonconvex semidefinite program. Remarkably, this nonconvex program is shown to admit a closed form solution, which may be of independent interest too. We complement this result by also studying a nonconvex Euclidean Frank-Wolfe approach, along with its global convergence analysis. Finally, we empirically compare RFW against recently published methods for the Riemannian centroid, and observe strong performance gains.
Preprint, 2017.

Traditionally, network analysis is based on local properties of ver- tices, like their degree or clustering, and their statistical behavior across the network in question. This paper develops an approach which is different in two respects: We investigate edge-based properties, and we define global characteristics of networks directly. More concretely, we start with Forman’s notion of the Ricci curvature of a graph, or more generally, a polyhedral complex. This will allow us to pass from a graph as representing a network to a polyhedral complex for instance by filling in triangles into connected triples of edges and to investigate the resulting effect on the curvature. This is insightful for two reasons: First, we can define a curvature flow in order to asymptotically simplify a network and reduce it to its essentials. Second, using a construction of Bloch which yields a discrete Gauß-Bonnet theorem, we have the Euler characteristic of a network as a global characteristic. These two aspects beautifully merge in the sense that the asymptotic properties of the curvature flow are indicated by that Euler characteristic.
In J. Compl. Net., 2017.

Recent Publications

More Publications

. Discrete Curvatures and Network Analysis. In MATCH, 2018.

PDF Project

. Heuristic Framework for Testing a Multi-Manifold Hypothesis. In Proc. of AWM: Research in Data Science, 2017.

PDF Code Project

Talks & Presentations

More Talks

Riemannian Frank-Wolfe with Application to the Geometric Matrix Mean

May 15, 2018, Bridging Mathematical Optimization, Information Theory and Data Science

Curvature-based Analysis of Complex Network

Apr 4, 2018, Seminar, Max Planck Institute for Mathematics in the Sciences

Curvature-based Analysis of Complex Network

Mar 28, 2018, EPFL Applied Topology Seminar

Projects

Curvature-based Network Analysis

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

Event Classification with CNNs

Convolutional Neural Networks for Event Classification at LHC’s ATLAS experiment.

Functional Analysis of Gene Networks

Efficient tools for functional characterization of gene networks.

Geometric Aspects in Machine Learing

Understanding and Learning the Geometry of Data.

Modeling Brain Disorders with Hopfield Networks

Neural networks model for effects of brain disorders on associative memory.

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