Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces

Mollenhauer, Mattes; Schuster, Ingmar; Klus, Stefan; Schütte, Christof

doi:10.1007/978-3-030-51264-4_5

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…(a) By sampling points from the uniform distribution, the Mercer feature map 35,36,42 with respect to the Lebesgue measure on X can be approximated by computing eigenfunctions of C XX -i.e., B = 1 n I and the auxiliary matrix eigenvalue problem is 1 n G XX v = λ v-as shown in Ref. 51. This can be easily extended to other measures.…”

Section: F Applications Of Rkhs Operatorsmentioning

confidence: 88%

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Kernel methods for detecting coherent structures in dynamical data

Klus

Husic

Mollenhauer

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that it can be obtained by optimizing the variational approach for Markov processes (VAMP) score. As a result, we show that coherent sets of particle trajectories can be computed by kernel CCA. We demonstrate the efficiency of this approach with several examples, namely the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Finally, we propose a straightforward generalization of dynamic mode decomposition (DMD) called coherent mode decomposition (CMD). Our results provide a generic machine learning approach to the computation of coherent sets with an objective score that can be used for cross-validation and the comparison of different methods.While coherent sets of particles are common in dynamical systems, they are notoriously challenging to identify. In this article, we leverage the combination of a suite of methods designed to approximate the eigenfunctions of transfer operators with kernel embeddings in order to design an algorithm for detecting coherent structures in Langrangian data. It turns out that the resulting method is a well-known technique to analyze relationships between multidimensional variables, namely kernel canonical correlation analysis. Our algorithm successfully identifies coherent structures in several diverse examples, including oceanic currents and a molecular dynamics problem with a moving potential. Furthermore, we show that a natural extension of our algorithm leads to a coherent mode decomposition, a counterpart to dynamic mode decomposition.

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Section: F Applications Of Rkhs Operatorsmentioning

confidence: 88%

“…A more detailed version of Proposition II.5 and its extension to the singular value decomposition are described in Ref. 51. Further properties of S and its decompositions will be studied in future work.…”

Section: E Empirical Rkhs Operatorsmentioning

confidence: 99%

Kernel methods for detecting coherent structures in dynamical data

Klus

Husic

Mollenhauer

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The SVD of empirical estimates of kernel transfer operators (and more general empirical RKHS operators) was recently proposed in Ref. 31 and might have additional applications such as low-rank approximation of operators or computing pseudoinverses of operators. Moreover, Gaussian processes might be beneficial to include also uncertainties in these algorithms.…”

Section: Discussionmentioning

confidence: 99%

A kernel-based approach to molecular conformation analysis

Klus

Bittracher

Schuster³

et al. 2018

The Journal of Chemical Physics

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We present a novel machine learning approach to understanding conformation dynamics of biomolecules. The approach combines kernel-based techniques that are popular in the machine learning community with transfer operator theory for analyzing dynamical systems in order to identify conformation dynamics based on molecular dynamics simulation data. We show that many of the prominent methods like Markov State Models, EDMD, and TICA can be regarded as special cases of this approach and that new efficient algorithms can be constructed based on this derivation. The results of these new powerful methods will be illustrated with several examples, in particular the alanine dipeptide and the protein NTL9.

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“…Mercer features of the Gaussian kernel and its antisymmetric and symmetric (see Section 4) counterparts-computed by a spectral decomposition of the covariance operator, cf. [19]are shown in Figure 2.…”

Section: Antisymmetric Kernelsmentioning

confidence: 99%

Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry

Klus,

Gelß,

Nüske

et al. 2021

Preprint

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We derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exploiting symmetries or antisymmetries the size of the training data set can be significantly reduced. The results are illustrated with guiding examples and simple quantum physics and chemistry applications.

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Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces

Cited by 3 publications

References 21 publications

Kernel methods for detecting coherent structures in dynamical data

Kernel methods for detecting coherent structures in dynamical data

A kernel-based approach to molecular conformation analysis

Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry

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