Transfer operators from optimal transport plans for coherent set detection

Koltai, Péter; Lindheim, Johannes von; Neumayer, Sebastian; Steidl, Gabriele

doi:10.1016/j.physd.2021.132980

Cited by 8 publications

(15 citation statements)

References 43 publications

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“…In particular its entropy regularized variant is rather popular in data analysis and machine learning applications [24]. It was recently proposed for the detection of coherent sets from data [18].…”

Section: Related Workmentioning

confidence: 99%

Entropic transfer operators

Junge¹,

Matthes²,

Schmitzer³

2022

Preprint

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We propose a new concept for the regularization and discretization of transfer operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer operator which can be analysed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n-torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

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Section: Related Workmentioning

confidence: 99%

Entropic transfer operators

Junge¹,

Matthes²,

Schmitzer³

2022

Preprint

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“…Moreover, the regularization readily introduces a diffusion into the dynamics, which is an important ingredient to solve the associated partitioning problem outlined below. For more details, we refer to (Koltai et al, 2021). In the case n > 2, we use multi-marginal unbalanced OT, see (Beier et al, 2021).…”

Section: Coherently Evolving Features: Dynamical Spectral Clusteringmentioning

confidence: 99%

“…. , n. If desired-e.g., for obtaining a clustering with more than two clusters using k-means, see, e.g., (von Luxburg, 2007;Koltai et al, 2021)-, further orthogonal partition functions can be obtained from singular vector pairs corresponding to smaller singular values σ j , j > 2.…”

Section: Coherently Evolving Features: Dynamical Spectral Clusteringmentioning

confidence: 99%

“…The name "Dynamical Spectral Clustering" is justified as follows: In the limiting case of a small regularization parameter ε of the OT, L * L is well approximated by the identity operator plus ε∆ dyn introduced in the previous section, see (Froyland, 2015;Koltai et al, 2021). This operator is symmetric, such that the singular decomposition above now becomes an eigendecomposition.…”

Section: Coherently Evolving Features: Dynamical Spectral Clusteringmentioning

confidence: 99%

“…The second part of the definition involves the persistence or coherence of the structures. In this context we will discuss the use of feature tracking approaches to directly estimate the lifetime of a structure (Günther et al, 2017;Sakurai et al, 2017;Heinze et al, 2017), Lagrangian coherent set analysis which provides estimates of the diffusive mixing time scale of a given set of Lagrangian trajectories (Froyland, 2013(Froyland, , 2015Banisch and Koltai, 2017;Koltai and Renger, 2018;Froyland and Koltai, 2021), and a new approach called "dynamic spectral clustering", which applies key ideas from the Lagrangian coherent set approach, but applies them to surrogate dynamics that are learned from the data and that may or may not approximate any Lagrangian paths (Koltai et al, 2021).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Definition, detection, and tracking of persistent structures in atmospheric flows

Lindheim¹,

Harikrishnan²,

Dörffel³

et al. 2021

Preprint

Self Cite

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Long-lived flow patterns in the atmosphere such as weather fronts, mid-latitude blockings or tropical cyclones often induce extreme weather conditions. As a consequence, their description, detection, and tracking has received increasing attention in recent years. Similar objectives also arise in diverse fields such as turbulence and combustion research, image analysis, and medical diagnostics under the headlines of "feature tracking", "coherent structure detection" or "image registration" -to name just a few. A host of different approaches to addressing the underlying, often very similar, tasks have been developed and successfully used. Here, several typical examples of such approaches are summarized, further developed and applied to meteorological data sets. Common abstract operational steps form the basis for a unifying framework for the specification of "persistent structures" involving the definition of the physical state of a system, the features of interest, and means of measuring their persistence.

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Gromov–Wasserstein Transfer Operators

Beier

2023

Lecture Notes in Computer Science

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Transfer operators from optimal transport plans for coherent set detection

Cited by 8 publications

References 43 publications

Entropic transfer operators

Entropic transfer operators

Definition, detection, and tracking of persistent structures in atmospheric flows

Gromov–Wasserstein Transfer Operators

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