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Data Classification Algorithms and Tverberg-type theorems

Banff International Research Station for Mathematical Innovation and Discovery

The classical Tverberg's theorem says that a set with sufficiently many points in $R^d$ can always be partitioned into m parts so that the (m - 1)-simplex is the (nerve) intersection pattern of the convex hulls of the parts. Motivated by questions from the performance of data classification algorithms such as multi-class logistic regression method, we investigated other version Tverberg. Our main results demonstrate that Tverberg's theorem is but a special case of a much more general situation. Given sufficiently many points, any tree or cycle, can also be induced by at least one partition of the point set. The proofs require a deep investigation of oriented matroids and order types. We also present new results on the distribution of simplicial complexes arising from the classification of data. (Joint work with Deborah Oliveros, Tommy Hogan, Dominic Yang (supported by NSF).)

Mathematics; Geometry; Convex and discrete geometry; Discrete mathematics

video/mp4

Author affiliation: University of California

2020-04-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73902

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/