@conference {5543,
	title = {Estimating Koopman operators with sketching to provably learn large scale dynamical systems},
	booktitle = {37th Conference on Neural Information Processing Systems (NeurIPS 2023)},
	year = {2023},
	month = {12/2023},
	abstract = {<p>The theory of Koopman operators allows to deploy non-parametric machine learn- ing algorithms to predict and analyze complex dynamical systems. Estimators such as principal component regression (PCR) or reduced rank regression (RRR) in ker- nel spaces can be shown to provably learn Koopman operators from finite empirical observations of the system{\textquoteright}s time evolution. Scaling these approaches to very long trajectories is a challenge and requires introducing suitable approximations to make computations feasible. In this paper, we boost the efficiency of different kernel- based Koopman operator estimators using random projections (sketching). We de- rive, implement and test the new {\textquotedblleft}sketched{\textquotedblright} estimators with extensive experiments on synthetic and large-scale molecular dynamics datasets. Further, we establish non asymptotic error bounds giving a sharp characterization of the trade-offs between statistical learning rates and computational efficiency. Our empirical and theoretical analysis shows that the proposed estimators provide a sound and efficient way to learn large scale dynamical systems. In particular our experiments indicate that the proposed estimators retain the same accuracy of PCR or RRR, while being much faster. Code is available at https://github.com/Giodiro/NystromKoopman.</p>
},
	url = {https://proceedings.neurips.cc/paper_files/paper/2023/file/f3d1e34a15c0af0954ae36a7f811c754-Paper-Conference.pdf},
	author = {Giacomo Meanti* and Antoine Chatalic and Vladimir R. Kostic and Pietro Novelli and Massimiliano Pontil and Lorenzo Rosasco}
}
@conference {5540,
	title = {Heteroscedastic Gaussian Processes and Random Features: Scalable Motion Primitives with Guarantees},
	booktitle = {7th Conference on Robot Learning (CoRL 2023},
	year = {2023},
	month = {11/2023},
	address = {Altanta, GA},
	abstract = {<p>Heteroscedastic Gaussian processes (HGPs) are kernel-based, non- parametric models that can be used to infer nonlinear functions with time-varying noise. In robotics, they can be employed for learning from demonstration as mo- tion primitives, i.e. as a model of the trajectories to be executed by the robot. HGPs provide variance estimates around the reference signal modeling the tra- jectory, capturing both the predictive uncertainty and the motion variability. How- ever, similarly to standard Gaussian processes they suffer from a cubic complexity in the number of training points, due to the inversion of the kernel matrix. The un- certainty can be leveraged for more complex learning tasks, such as inferring the variable impedance profile required from a robotic manipulator. However, suit- able approximations are needed to make HGPs scalable, at the price of potentially worsening the posterior mean and variance profiles. Motivated by these obser- vations, we study the combination of HGPs and random features, which are a popular, data-independent approximation strategy of kernel functions. In a the- oretical analysis, we provide novel guarantees on the approximation error of the HGP posterior due to random features. Moreover, we validate this scalable motion primitive on real robot data, related to the problem of variable impedance learning. In this way, we show that random features offer a viable and theoretically sound alternative for speeding up the trajectory processing, without sacrificing accuracy.</p>
},
	url = {https://proceedings.mlr.press/v229/caldarelli23a/caldarelli23a.pdf},
	author = {Edoardo Caldarelli and Antoine Chatalic and Adri{\textquoteleft}a Colom{\textasciiacute}e and Lorenzo Rosasco and Carme Torras}
}
