Spectral Pollution in Kernel DMD: What Projection Can Change

This post is a continuation of:

Eigenvalue Sensitivity: Pseudospectra for Non-Normal Operators

Kernel DMD and EDMD estimate finite-dimensional descriptions of an operator acting on observables. Their eigenvalues can be useful, but a computed eigenpair need not be an eigenpair of the full Koopman operator.

Three issues need to be distinguished: sensitivity of a fixed matrix to perturbation, approximation of continuous spectrum by discrete spectra, and spectral pollution outside the true spectrum. Increasing dictionary size alone does not resolve any of them.

The Hidden Assumption

When we compute Koopman spectra from data, we implicitly assume:

Finite-rank approximations preserve the spectral structure of the true operator.

That assumption is false in general. The Koopman operator acts on an infinite-dimensional function space. Kernel EDMD replaces it with a Galerkin projection onto a finite subspace. This is not a benign truncation; it fundamentally changes the spectral problem.

Strong operator convergence alone does not ensure spectral convergence. Operator-norm convergence is stronger: for bounded operators on a common space, if in norm and converges to , then . This excludes spectral pollution in that setting, although it does not say that is in the point spectrum.

One way to see this is that invertibility persists under sufficiently small norm perturbations: a point in the resolvent set cannot be the limit of such spectral points. See Beckus, spectral-convergence notes, §5.5.

Finite-rank operators can converge in norm only to compact operators. Consequently, norm convergence is generally unavailable for approximating a noncompact Koopman operator, including a unitary operator on an infinite-dimensional space.

Resolvent contours and eigenvalues of a finite non-normal reference matrix — Figure 1 — Resolvent geometry for the finite reference matrix labeled . The black points are that matrix's eigenvalues; “true eigenvalues” in the legend refers to this reference matrix, not to a verified infinite-dimensional Koopman spectrum. The dashed curve is the unit circle.

What Compression Can Hide

Let be the Koopman operator and be the projection onto your -dimensional dictionary. The object you actually diagonalize is the compressed operator:

Here the restriction makes an operator on the finite-dimensional approximation space. Finite-data EDMD additionally estimates the inner products defining this idealized projection.

A small example shows what compression can hide. Take:

The compression to the span of is the scalar , although . Its projected eigenpair has zero residual in the compressed problem, while its full residual is .

This is a finite-dimensional illustration of spurious spectrum under compression, not by itself an asymptotic pollution example. It also shows that non-normality is unnecessary: is self-adjoint. Restoring the full two-dimensional space recovers both true eigenvalues.

What Spectral Pollution Actually Is

For an approximation sequence, spectral pollution means that some converges to a point . Such points can persist under refinement; visual stability does not prove correctness.

Spectral points at ranks 10, 20, 40, and 80 overlaid on resolvent contours — Figure 2 — The displayed spectra become denser as rank increases. Despite the title's “pollution progression” wording, density and proximity to resolvent contours alone do not establish spectral pollution: that requires limiting points outside the full operator's spectrum.

A different possibility is convergence to the continuous spectrum. That point belongs to the true spectrum but is not an eigenvalue. Interpreting it as a genuine Koopman eigenfunction would be mistaken, yet the spectral location is not pollution.

Schematic continuous spectral region compared with discrete eigenvalues of a finite approximation — Figure 3 — A schematic contrast between continuous spectrum and a finite matrix's point spectrum. The annulus is illustrative, not a universal Koopman spectrum; “no point spectrum” concerns the depicted component. Finite eigenvalues lying within continuous spectrum are not, solely for that reason, spectral pollution.

A useful precision statement is:

The limiting point might belong to continuous spectrum, or lie outside the spectrum altogether. The displayed implication concerns point spectrum and does not distinguish these cases.

Kernel DMD and the Choice of Function Space

Implicit Dictionaries: The RKHS feature space is high-dimensional, but the effective rank is controlled by data and regularization. This obscures the true approximation dimension.
Operator and space choice: Non-normality can amplify sensitivity, but is not universal. For invertible measure-preserving dynamics, the Koopman operator on the invariant-measure space is unitary. Its compression need not be normal.
Sampling-Driven Subspaces: The Galerkin subspace is spanned by data-dependent features. Change the data slightly, and the spectrum changes.

For a bounded normal operator on a Hilbert space, and outside its spectrum, the resolvent satisfies

For a non-normal operator, the resolvent norm can be large far from the spectrum. This explains sensitivity to small operator perturbations. It does not establish that arbitrary Galerkin eigenvalues follow resolvent contours: compression is not generally a small perturbation of the full operator.

For the role of invariant measures and function spaces, see Mezić, Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State Space Geometry.

Prediction and Spectral Interpretation

Regularization and dictionary size affect both prediction and the fitted spectrum. Increasing capacity can reduce approximation error while making estimation harder, but neither effect is universally monotone.

Two mode expansions can make similar short-horizon predictions while assigning different eigenvalues to their components. Predictive accuracy therefore answers a different question from whether an individual mode approximates a true Koopman eigenfunction.

Qualitative sketch of approximation bias, estimation variance, and spectral pollution risk versus model capacity — Figure 4 — A qualitative sketch of possible capacity tradeoffs, not measured error curves. The annotations “more capacity = more pollution” and “grows unbounded with capacity” are not general mathematical conclusions: pollution can persist, diminish, or be absent as the approximation space grows.

Residuals Beyond the Projection

For an orthogonal projection and a candidate in its range, the full eigenpair error splits as:

A matrix eigensolver can make the first term zero while the second remains large. This is precisely what happens in the two-dimensional example above. Recomputing that matrix's eigenvalues or resolvent does not recover the omitted term.

Colbrook, Ayton and Szőke's ResDMD paper explains how data can estimate residuals associated with the full Koopman operator. Its spectral verification results depend on the stated sampling and approximation assumptions. A small residual alone still does not distinguish an exact eigenfunction from an approximate one associated with continuous spectrum or non-normal sensitivity.

Sensitivity to changes in data, dictionary, or regularization remains useful diagnostic evidence. It is not a theorem that genuine modes must be insensitive, nor that stable computed modes must be genuine.

Eigenvalue locations and their movement under random matrix perturbations — Figure 5 — The plot contrasts eigenvalues with smaller and larger movement under the displayed matrix perturbations. “Stable” describes sensitivity in this test, not certified Koopman modes. These perturbations are not automatically equivalent to data subsampling or changes in kernel bandwidth, and movement alone does not classify spectral pollution.