We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d. settings, the latter in terms of mixing coefficients. Our results suggest RRR might be beneficial over other widely used estimators as confirmed in numerical experiments both for forecasting and mode decomposition.

The talks discusses work from the papers: https://arxiv.org/pdf/2205.14027.pdf

BIO: Massimiliano Pontil is Senior Reseacher at the Italian Institute of Technology (https://www.iit.it/people/massimiliano-pontil) and professor of Computational Statistics and Machine Learning at University College London. He is also co-director of the ELLIS Unit Genoa (https://ellisgenoa.eu/), a joint effort of IIT and University of Genoa, and a member of the UCL Centre for Artificial Intelligence (https://www.ucl.ac.uk/ai-centre/). He has been active in machine learning research for over twenty years, working on theory and algorithms, including the areas of kernel methods, multitask and transfer learning, online learning, sparsity regularisation, and statistical learning theory. Recent interests include meta-learning, algorithm fairness, and hyper-parameter optimization.