Nonlinear state-space models are powerful tools to describe dynamical structures in complex time series. In a streaming setting where data are processed one sample at a time, simultaneous inference of the state and its nonlinear dynamics has posed significant challenges in practice. We develop a novel online learning framework, leveraging variational inference and sequential Monte Carlo, which enables flexible and accurate Bayesian joint filtering. Our method provides an approximation of the filtering posterior which can be made arbitrarily close to the true filtering distribution for a wide class of dynamics models and observation models. Specifically, the proposed framework can efficiently approximate a posterior over the dynamics using sparse Gaussian processes, allowing for an interpretable model of the latent dynamics. Constant time complexity per sample makes our approach amenable to online learning scenarios and suitable for real-time applications.
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Sparse Learning of Dynamical Systems in RKHS: An Operator-Theoretic Approach
Transfer operators provide a rich framework for representing the dynamics of very general, nonlinear dynamical systems. When interacting with reproducing kernel Hilbert spaces (RKHS), descriptions of dynamics often incur prohibitive data storage requirements, motivating dataset sparsification as a precursory step to computation. Further, in practice, data is available in the form of trajectories, introducing correlation between samples. In this work, we present a method for sparse learning of transfer operators from $\beta$-mixing stochastic processes, in both discrete and continuous time, and provide sample complexity analysis extending existing theoretical guarantees for learning from non-sparse, i.i.d. data. In addressing continuous-time settings, we develop precise descriptions using covariance-type operators for the infinitesimal generator that aids in the sample complexity analysis. We empirically illustrate the efficacy of our sparse embedding approach through deterministic and stochastic nonlinear system examples.
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- Award ID(s):
- 2031570
- NSF-PAR ID:
- 10468368
- Editor(s):
- Krause, Andreas; Brunskill, Emma; Cho, Kyunghyun; Engelhardt, Barbara; Sabato, Sivan; Scarlett, Jonathan
- Publisher / Repository:
- Proceedings of the 40th International Conference on Machine Learning
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 202
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 13325--13352
- Format(s):
- Medium: X
- Location:
- Hawaii, USA
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
