Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces

1 Introduction

• Transfer-operator eigenfunctions can separate fast and slow dynamics, identify metastable sets, and support signal separation.The paper highlights molecular dynamics, video, and text as application settings.Source: §1 · p. 1
• The paper constructs RKHS representations of transfer operators using covariance and cross-covariance operators, without requiring explicitly defined basis functions.The framework supports infinite-dimensional feature spaces and domains such as images, graphs, and strings when an appropriate kernel is available.Source: §1 · p. 2
• The paper relates embedded Perron–Frobenius operators to conditional mean embeddings and proposes an eigendecomposition method for finite-rank RKHS operators.The authors also present kernel transfer operators and empirical estimators, and identify existing transfer-operator methods as special cases.Source: §1 · p. 2
• The framework is positioned for applications including molecular dynamics, video, text, and EEG data analysis.Potential uses include visualization, dimension reduction, source separation, denoising, data summarization, and sequence-based clustering.Source: §1 · p. 2

2 Notation and Preliminaries

• In the dynamical-system applications, X and Y denote states at times t and t + τ, with τ a fixed lag time.The section introduces notation and background needed to define and analyze kernel transfer operators.Source: §2 · p. 3

2.1 Reproducing Kernel Hilbert Spaces

• An RKHS represents points through kernel feature maps, so inner products and many learning computations can use finite Gram matrices without explicitly constructing features.Kernels may correspond to infinite-dimensional feature spaces, while Gram matrices contain pairwise kernel evaluations on data.Source: §2.1 · pp. 3–4

2.2 Hilbert Space Embedding of Distributions

• Kernel mean embeddings represent distributions in an RKHS, while conditional mean embeddings represent conditional distributions through covariance and cross-covariance operators.The conditional embedding is commonly regularized because the required conditional-expectation function may not belong to the input RKHS; characteristic kernels can fully characterize distributions.Source: §2.2 · pp. 5–7

2.3 Transfer Operators

• Transfer operators describe lagged evolution of densities or observables; for reversible systems their eigenvalues are real and their eigenfunctions form an orthogonal basis.The Ornstein–Uhlenbeck process is used as a guiding example, and Koopman eigenfunctions can support evaluation of a system at data points even when its explicit form is unknown.Source: §2.3 · pp. 8–9

3 Kernel Transfer Operators and Their Eigendecompositions

• The formulation assumes the lagged input and output occupy the same state space and use the same kernel and RKHS.It defines time-lagged Gram matrices to express the operators in terms of data.Source: §3 · p. 9
• The paper derives RKHS transfer operators for densities and observables, alongside operators that act on embedded densities and observables.The kernel Perron–Frobenius operator propagates densities directly in the RKHS, unlike the conditional mean embedding, which propagates embedded densities.Source: §3 · p. 9
• The kernel Perron–Frobenius operator is presented as a direct RKHS representation of density propagation, distinct from propagation only in embedded form.This distinction is highlighted as particularly relevant to the RKHS perspective.Source: §3 · p. 9

3.1 Main Results

• The four kernel transfer operators include direct density and observable propagation, which requires the relevant densities and observables to lie in an invariant RKHS subspace.The paper characterizes this as a strong assumption that may fail for more complex chaotic systems; embedded operators instead propagate kernel mean embeddings or embedded observables without that membership requirement.Source: §3.1 · p. 9, Table 2 · p. 10
• The empirical estimates for embedded operators converge in operator norm at order O_p(…), with ε denoting a regularization parameter.The supplied passage states the rate only as an incomplete expression; it does not support a more specific rate.Source: §3.1 · p. 10
• A general finite-rank RKHS operator’s eigendecomposition can be computed by solving a real-valued surrogate problem based on Gram matrices.The resulting operator eigenfunctions are summarized alongside the surrogate eigenproblems.Source: §3.1 · p. 10, Table 3 · p. 10

3.2 Eigendecomposition of RKHS Operators

• Nonzero eigenvalues of S = ΥBΓ⊤ and its eigenfunctions can be recovered from eigenvectors of finite matrices BGΓΥ or GΓΥB.The corresponding eigenfunctions are Υv and ΓG−1ΓΓv, respectively.Source: §3.2 · p. 10
• The eigendecomposition expressions cover transfer-operator estimators, and include KPCA as a special case extended to more general RKHS operators.For explicitly known finite-dimensional feature spaces, the matrix size does not depend on the number of test points.Source: §3.2 · pp. 10–11
• For the Ornstein–Uhlenbeck example, the dominant numerical eigenfunctions of the kernel Perron–Frobenius and Koopman operators are in good agreement with analytical results.Figure 1 distinguishes numerical solid curves from analytical dotted curves in separate panels for the two operators.Source: §3.2 · p. 11, Figure 1 · p. 11

3.3.1 Kernel Perron–Frobenius Operator

• The kernel Perron–Frobenius operator propagates densities directly in the RKHS, assuming the relevant densities belong to that space.In general they may not, and the paper asks under what conditions this operator approximates the L1 Perron–Frobenius operator.Source: §3.3 · p. 11, §3.3.1 · p. 12
• The empirical kernel Perron–Frobenius estimate has the form bPk = ΨAΦ⊤, with A specified by a regularized Gram-matrix expression.The operator can also be expressed using covariance and cross-covariance operators; its definition requires matching domain and feature spaces.Source: §3.3.1 · pp. 12–13

3.3.2 Embedded Perron–Frobenius Operator

• The embedded Perron–Frobenius operator is the conditional mean embedding, mapping embedded densities forward; embedding a Perron–Frobenius eigenfunction yields an eigenfunction of this operator.The correspondence requires the stated existence conditions for the conditional mean embedding and the density embedding.Source: §3.3.2 · pp. 13–15
• Whether embedded eigenfunctions are easier to approximate or can directly identify metastable states and lower-dimensional representations remains open.The paper notes that using preimages of embedded functions may instead be necessary.Source: §3.3.2 · p. 15

3.4.1 Kernel Koopman Operator

• The kernel Koopman operator acts on observables in the RKHS and is expressed as Kk = (CXX + εI)−1CXY.Its empirical estimate is introduced using sampled state transitions.Source: §3.4.1 · p. 16
• For the example's finite feature space, the computed dominant eigenvalues and eigenfunctions agree well with analytical results, but products beyond ϕ4 are not representable in the RKHS.The restriction follows because the feature space contains no polynomials of order greater than two.Source: §3.4.1 · p. 16

3.4.2 Embedded Koopman Operator

• The embedded Koopman operator maps embedded observables forward, and embedding an eigenfunction of K yields an eigenfunction of KE.The empirical operator estimate is defined using covariance and cross-covariance operators.Source: §3.4.2 · p. 17
• In the example, the vectors associated with the embedded eigenfunctions are eigenvectors of the empirical embedded Koopman operator when ε = 0.The vectors correspond to embedded eigenfunctions computed analytically from the example system.Source: §3.4.2 · p. 17

3.5 Consistency and Convergence Rate

• The section bounds estimation error but leaves a complete convergence analysis of the kernel transfer operators to future work.Source: §3.5 · p. 18
• With ε = n^-α for α < 1/2, the stated overall rate is n^(α-1/2), and ε must decay for approximation error to vanish.The authors leave detailed analysis of the approximation error to future work.Source: §3.5 · p. 19
• Unlike cited conditional mean embedding results, this result concerns convergence in operator norm rather than RKHS norm.Source: §3.5 · p. 19
• Theorem 3.14 establishes consistent empirical estimation at a reasonably fast rate without parametric assumptions on the underlying dynamical system.The result assumes a separable RKHS with a measurable kernel and finite expected kernel diagonals at both time points.Source: §3.5 · pp. 18–19

3.6 Relationships with Other Methods

• The RKHS approach provides a unified framework that subsumes existing transfer-operator approximation methods, including TICA, DMD, VAC, and EDMD.Source: §3.6 · p. 19

3.6.1 TICA and DMD

• TICA and DMD are recovered as special cases; TICA coordinates capture time-lagged components, while DMD modes correspond to coherent flow structures.Source: §3.6.1 · pp. 19–20

3.6.2 VAC and EDMD

• VAC and EDMD use explicitly specified feature bases and are equivalent for reversible systems, acting as nonlinear extensions of TICA and DMD, respectively.Source: §3.6.2 · p. 20

3.6.3 Kernel TICA and Kernel EDMD

• Kernel TICA and kernel EDMD express the eigenvalue problem through Gram matrices, avoiding explicit transformed feature matrices and allowing implicit infinite-dimensional feature spaces.Source: §3.6.3 · p. 20

3.6.4 Conditional Mean Embedding

• The embedded Perron–Frobenius operator has the same form as the conditional mean embedding of the next-state distribution given the current state; it is adjoint to Kk.Source: §3.6.4 · p. 21

4.1 Molecular Dynamics

• Five dominant Koopman eigenfunctions partition the multi-well diffusion domain into five metastable sets corresponding to the potential wells.The eigenvalues show a gap between the fifth and sixth.Source: §4.1 · p. 22, Figure 3 · p. 22
• TICA reduces a four-variable time series to two metastable coordinates, while the other two variables contain noise.The first two TICA eigenvalues are close to 1 and the remaining two are close to 0.Source: §4.1 · pp. 22–23, Figure 4 · p. 23
• For n-butane, eigenfunctions computed from the 12-dimensional carbon-atom state space parametrize the dihedral angle and identify three metastable sets.The sets occur around ϑ = 60°, 180°, and approximately 300°; their configurations depend mainly on the dihedral angle.Source: §4.1 · pp. 23–24, Figure 6 · p. 24, Figure 7 · p. 25

4.2 Movie Data

• Pendulum eigenfunctions encode its frequency and twice that frequency, allowing frames to be sorted by angular displacement.The dominant eigenfunction parametrizes the angular displacement.Source: §4.2 · p. 25, Figure 8 · p. 26

4.3 Text Data

• The text example treats successive vocabulary words in a sequence of news articles as state-to-successor transitions.The article sequence can produce rare transitions between topics or between different uses of a word.Source: §4.3 · pp. 25, 27
• Eigenfunction-based clustering identifies four topic groups: information technology, medicine, weather, and politics.Words cluster by proximity in the document collection, rather than by string-kernel similarity.Source: Figure 9 · p. 27, §4.3 · p. 28
• The text kernel captures shared substrings, including noncontiguous matches, so clusters can reflect grammatical variants and misspellings without normalization.The passage contrasts this with a pure Markov state model, which does not account for string similarity.Source: §4.3 · p. 28

• Kernel transfer operators propagate embedded densities and connect the covariance-operator approach with TICA and EDMD for approximating transfer operators.Source: §5 · p. 28
• The framework applies to non-vectorial data such as strings and graphs, and is demonstrated with molecular-dynamics, video, and text examples.Source: §5 · p. 28
• Convergence of kernel transfer operators to the actual transfer operators remains the main theoretical question.Source: §5 · p. 28
• Applying the methods to more realistic datasets, especially more complicated video, is future work.Source: §5 · p. 28
• The influence of test-point count on eigenfunction-approximation accuracy is not yet clear.Source: §5 · p. 29
• The authors propose singular-value decompositions as an extension for problems with different input and output spaces.Source: §5 · p. 29