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Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport
Theo Lacombe · Marco Cuturi · Steve OUDOT

Thu Dec 06 02:00 PM -- 04:00 PM (PST) @ Room 210 #72

Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. Indeed, this was recently exemplified in a string of papers which show that the simple task of averaging a few PDs can be computationally prohibitive. We propose in this article a tractable framework to carry out standard tasks on PDs at scale, notably evaluating distances, estimating barycenters and performing clustering. This framework builds upon a reformulation of PD metrics as optimal transport (OT) problems. Doing so, we can exploit recent computational advances: the OT problem on a planar grid, when regularized with entropy, is convex can be solved in linear time using the Sinkhorn algorithm and convolutions. This results in scalable computations that can stream on GPUs. We demonstrate the efficiency of our approach by carrying out clustering with diagrams metrics on several thousands of PDs, a scale never seen before in the literature.

Author Information

Theo Lacombe (Inria Saclay)
Marco Cuturi (Google Brain & CREST - ENSAE)

Marco Cuturi is a research scientist at Apple, in Paris. He received his Ph.D. in 11/2005 from the Ecole des Mines de Paris in applied mathematics. Before that he graduated from National School of Statistics (ENSAE) with a master degree (MVA) from ENS Cachan. He worked as a post-doctoral researcher at the Institute of Statistical Mathematics, Tokyo, between 11/2005 and 3/2007 and then in the financial industry between 4/2007 and 9/2008. After working at the ORFE department of Princeton University as a lecturer between 2/2009 and 8/2010, he was at the Graduate School of Informatics of Kyoto University between 9/2010 and 9/2016 as a tenured associate professor. He joined ENSAE in 9/2016 as a professor, where he is now working part-time. He was at Google between 10/2018 and 1/2022. His main employment is now with Apple, since 1/2022, as a research scientist working on fundamental aspects of machine learning.

Steve OUDOT (inria)

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