The thirty-eighth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, December 2, 2021.
2–3 PM — Luc Pronzato (CNRS, Univ. Côte d'Azur) — [slides]
Maximum Mean Discrepancy, Bayesian integration and kernel herding for space-filling design
A standard objective in computer experiments is to predict/interpolate the behaviour of an unknown function f on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, minimax-distance) or a discrepancy criterion measuring distance to uniformity.
We focus our attention to sequential constructions where design points are added one at a time. The presentation is based on the survey [4], built on several recent results [2, 5, 6] that show how energy functionals can be used to measure distance to uniformity. We investigate connections between design for integration of f with respect to a measure µ (quadrature design), construction of the (continuous) BLUE for the location model, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative showing the strong relation between discrepancy minimization and more traditional design of optimal experiments, as used for instance in [3]. Kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design [1, 7], can be applied to the construction of point sequences with suitable space-filling properties. Several illustrative examples are presented
Refs:
- F. Bach, S. Lacoste-Julien, and G. Obozinski. On the equivalence between herding and conditional gradient algorithms. In Proc. 29th Annual International Conference on Machine Learning, pages 1355–1362, 2012.
- S.B. Damelin, F.J. Hickernell, D.L. Ragozin, and X. Zeng. On energy, discrepancy and group invariant measures on measurable subsets of Euclidean space. J. Fourier Anal. Appl., 16:813–839, 2010.
- S. Mak and V.R. Joseph. Support points. Annals of Statistics, 46(6A):2562–2592, 2018.
- L. Pronzato and A.A. Zhigljavsky. Bayesian quadrature, energy minimization and space-filling design. SIAM/ASA J. Uncertainty Quantification, 8(3):959–1011, 2020.
- S. Sejdinovic, B. Sriperumbudur, A. Gretton, and K. Fukumizu. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. The Annals of Statistics, 41(5):2263–2291, 2013.
- B.K. Sriperumbudur, A. Gretton, K. Fukumizu, B. Schölkopf, and G.R.G. Lanckriet. Hilbert space embeddings and metrics on probability measures. Journal of Machine Learning Research, 11:1517–1561, 2010.
- M. Welling. Herding dynamical weights to learn. In Proc. 26th Annual International Conference on Machine Learning, pages 1121–1128, 2009.
Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Didier Clouteau (MSSMAT), Amélie Fau (LMT), Filippo Gatti (MSSMAT), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (DOTA), Fernando Lopez-Caballero (MSSMAT), Didier Lucor (LISN), Emmanuel Vazquez (L2S).
Coordinator: Julien Bect (L2S).
Practical details: the seminar will be held online using Microsoft Teams.
If you want to attend this seminar (or any of the forthcoming online UQSay seminars), and if you do not already have access to the UQSay group on Teams, simply send an email and you will be invited. Please specify which email address the invitation must be sent to (this has to be the address associated with your Teams account).
You will find the link to the seminar on the "General" UQSay channel on Teams, approximately 15 minutes before the beginning.
The technical side of things: you can use Teams either directly from your web browser or using the "fat client", which is available for most platforms (Windows, Linux, Mac, Android & iOS). We strongly recommend the latter option whenever possible. Please give it a try before the seminar to anticipate potential problems.
