Invited Speakers

Yue Ren, Swansea University

Ximena Fernandez, Swansea University

Date:

15 December 2020

Location:

Department of Mathematics, University of York, on Remo and Zoom (details will be sent to the AAG mailing list)

Local organiser

Emilie Dufresne, University of York

Schedule:

• 13:30-14:00 Welcome (on Remo)

• 14:00-14:45 Ximena Fernandez: Geometric and Topological Inference for Data Analysis (on Zoom)

• 14:45-15:15 Break (on Remo)

• 15:15-16:00 Yue Ren: Tropical varieties of neural networks (on Zoom)

Titles and Abstracts

Ximena Fernandez: Geometric and Topological Inference for Data Analysis.

Abstract: In this talk we approach the problem of learning information about a geometric object from a finite set of (possibly noisy) sample points drawn respect to some unknown distribution. More concretely, given a smooth manifold and a density that produces the sample, we consider an intrinsic density-based metric, known as the Fermat distance. We construct a computable distance over the sample and prove that this sample metric space is a good estimator of the manifold (in the sense of Gromov-Haussdorf). Finally, we present some applications of this result in topological data analysis, showing how this approach outperforms more standard methods with computational experiments in synthetic and real datasets.

Yue Ren: Tropical varieties of neural networks.

Abstract: In this talk, we introduce tropical varieties arising from neural networks with piecewise linear activations, and discuss how their geometry affects their expressivity. In particular, we will use Weibel's f-Vector Theorem to derive optimal bounds for single-layered maxout networks, and Speyer's f-Vector Theorem to analyse networks with heavily restricted weights. We conclude with an initializing strategy for maxout networks based on our results.

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Sponsors:

We are grateful for the financial support from the Glasgow Mathematical Journal Learning and Research Support Fund, from the Edinburgh Mathematical Society, the London Mathematical Society.