Spatiotemporal Feature Extraction with Data-Driven Koopman Operators

Dimitrios Giannakis, Joanna Slawinska, Zhizhen Zhao
Proceedings of the 1st International Workshop on Feature Extraction: Modern Questions and Challenges at NIPS 2015, PMLR 44:103-115, 2015.

Abstract

We present a framework for feature extraction and mode decomposition of spatiotemporal data generated by ergodic dynamical systems. Unlike feature extraction techniques based on kernel operators, our approach is to construct feature maps using eigenfunctions of the Koopman group of unitary operators governing the dynamical evolution of observables and probability measures. We compute the eigenvalues and eigenfunctions of the Koopman group through a Galerkin scheme applied to time-ordered data without requiring a priori knowledge of the dynamical evolution equations. This scheme employs a data-driven set of basis functions on the state space manifold, computed through the diffusion maps algorithm and a variable-bandwidth kernel designed to enforce orthogonality with respect to the invariant measure of the dynamics. The features extracted via this approach have strong timescale separation, favorable predictability properties, and high smoothness on the state space manifold. The extracted features are also invariant under weakly restrictive changes of observation modality. We apply this scheme to a synthetic dataset featuring superimposed traveling waves in a one-dimensional periodic domain and satellite observations of organized convection in the tropical atmosphere.

Cite this Paper

BibTeX
@InProceedings{pmlr-v44-giannakis15, title = {Spatiotemporal Feature Extraction with Data-Driven Koopman Operators}, author = {Giannakis, Dimitrios and Slawinska, Joanna and Zhao, Zhizhen}, booktitle = {Proceedings of the 1st International Workshop on Feature Extraction: Modern Questions and Challenges at NIPS 2015}, pages = {103--115}, year = {2015}, editor = {Storcheus, Dmitry and Rostamizadeh, Afshin and Kumar, Sanjiv}, volume = {44}, series = {Proceedings of Machine Learning Research}, address = {Montreal, Canada}, month = {11 Dec}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v44/giannakis15.pdf}, url = {https://proceedings.mlr.press/v44/giannakis15.html}, abstract = {We present a framework for feature extraction and mode decomposition of spatiotemporal data generated by ergodic dynamical systems. Unlike feature extraction techniques based on kernel operators, our approach is to construct feature maps using eigenfunctions of the Koopman group of unitary operators governing the dynamical evolution of observables and probability measures. We compute the eigenvalues and eigenfunctions of the Koopman group through a Galerkin scheme applied to time-ordered data without requiring a priori knowledge of the dynamical evolution equations. This scheme employs a data-driven set of basis functions on the state space manifold, computed through the diffusion maps algorithm and a variable-bandwidth kernel designed to enforce orthogonality with respect to the invariant measure of the dynamics. The features extracted via this approach have strong timescale separation, favorable predictability properties, and high smoothness on the state space manifold. The extracted features are also invariant under weakly restrictive changes of observation modality. We apply this scheme to a synthetic dataset featuring superimposed traveling waves in a one-dimensional periodic domain and satellite observations of organized convection in the tropical atmosphere.} }
Endnote
%0 Conference Paper %T Spatiotemporal Feature Extraction with Data-Driven Koopman Operators %A Dimitrios Giannakis %A Joanna Slawinska %A Zhizhen Zhao %B Proceedings of the 1st International Workshop on Feature Extraction: Modern Questions and Challenges at NIPS 2015 %C Proceedings of Machine Learning Research %D 2015 %E Dmitry Storcheus %E Afshin Rostamizadeh %E Sanjiv Kumar %F pmlr-v44-giannakis15 %I PMLR %P 103--115 %U https://proceedings.mlr.press/v44/giannakis15.html %V 44 %X We present a framework for feature extraction and mode decomposition of spatiotemporal data generated by ergodic dynamical systems. Unlike feature extraction techniques based on kernel operators, our approach is to construct feature maps using eigenfunctions of the Koopman group of unitary operators governing the dynamical evolution of observables and probability measures. We compute the eigenvalues and eigenfunctions of the Koopman group through a Galerkin scheme applied to time-ordered data without requiring a priori knowledge of the dynamical evolution equations. This scheme employs a data-driven set of basis functions on the state space manifold, computed through the diffusion maps algorithm and a variable-bandwidth kernel designed to enforce orthogonality with respect to the invariant measure of the dynamics. The features extracted via this approach have strong timescale separation, favorable predictability properties, and high smoothness on the state space manifold. The extracted features are also invariant under weakly restrictive changes of observation modality. We apply this scheme to a synthetic dataset featuring superimposed traveling waves in a one-dimensional periodic domain and satellite observations of organized convection in the tropical atmosphere.
RIS
TY - CPAPER TI - Spatiotemporal Feature Extraction with Data-Driven Koopman Operators AU - Dimitrios Giannakis AU - Joanna Slawinska AU - Zhizhen Zhao BT - Proceedings of the 1st International Workshop on Feature Extraction: Modern Questions and Challenges at NIPS 2015 DA - 2015/12/08 ED - Dmitry Storcheus ED - Afshin Rostamizadeh ED - Sanjiv Kumar ID - pmlr-v44-giannakis15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 44 SP - 103 EP - 115 L1 - http://proceedings.mlr.press/v44/giannakis15.pdf UR - https://proceedings.mlr.press/v44/giannakis15.html AB - We present a framework for feature extraction and mode decomposition of spatiotemporal data generated by ergodic dynamical systems. Unlike feature extraction techniques based on kernel operators, our approach is to construct feature maps using eigenfunctions of the Koopman group of unitary operators governing the dynamical evolution of observables and probability measures. We compute the eigenvalues and eigenfunctions of the Koopman group through a Galerkin scheme applied to time-ordered data without requiring a priori knowledge of the dynamical evolution equations. This scheme employs a data-driven set of basis functions on the state space manifold, computed through the diffusion maps algorithm and a variable-bandwidth kernel designed to enforce orthogonality with respect to the invariant measure of the dynamics. The features extracted via this approach have strong timescale separation, favorable predictability properties, and high smoothness on the state space manifold. The extracted features are also invariant under weakly restrictive changes of observation modality. We apply this scheme to a synthetic dataset featuring superimposed traveling waves in a one-dimensional periodic domain and satellite observations of organized convection in the tropical atmosphere. ER -
APA
Giannakis, D., Slawinska, J. & Zhao, Z.. (2015). Spatiotemporal Feature Extraction with Data-Driven Koopman Operators. Proceedings of the 1st International Workshop on Feature Extraction: Modern Questions and Challenges at NIPS 2015, in Proceedings of Machine Learning Research 44:103-115 Available from https://proceedings.mlr.press/v44/giannakis15.html.

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