Selected Publications

We consider the problem of learning representations of relational data in spaces of constant sectional curvature, i.e., Euclidean, Hyperbolic, and Spherical space. In this context, we explore how to identify a suitable embedding curvature for a given relational dataset. For this task, we investigate the use of a scalable heuristic based on local graph neighborhoods and evaluate it on classic benchmark graphs.
In NIPS R2L, 2018.

We study projection free methods for constrained geodesically convex optimization. In particular, we propose a Riemannian version of the Frank-Wolfe (RFW) method and provide global, non-asymptotic sublinear (and, in a special case, linear) convergence guarantees. Later, we specialize RFW to the manifold of positive definite matrices, motivated by the specific task of computing the geometric mean (or Karcher mean/ Riemannian centroid). For this, 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. 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 vertices, like their degree or clustering, and their statistical behavior across the network in question. Here, we investigate edge-based properties and define global characteristics of networks directly. We start with Forman’s notion of the Ricci curvature of a graph allowing us to pass from a graph as representing a network to a polyhedral complex and to investigate the resulting effect on the curvature. With this, we can define a curvature flow in order to asymptotically simplify a network and reduce it to its essentials. Using a construction of Bloch which yields a discrete Gauß-Bonnet theorem, we define the Euler characteristic of a network as a global characteristic and the asymptotic properties of the curvature flow are indicated by that Euler characteristic.
In J. Compl. Net., 2017.

Talks & Presentations

More Talks

Curvature-based Analysis of Complex Network

Aug 28, 2018, NYC Machine Learning and Artificial Intelligence Meetup

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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