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Interactive learning studio · 6 lessons

The Geometry of
Transient Amplification

From stable eigenvalues to kernel-induced representation growth

\[\text{Dynamics}\longrightarrow\text{RKHS geometry}\longrightarrow\text{Kreiss sensitivity}\longrightarrow\text{Transient growth}\]

Himanshu Singh · NeuReps 2026 Proceedings Track

Use → / ← to advance, ↓ to explore a lesson, F for fullscreen, O for slide overview. Swipe on touchscreens.

Course map

Six questions guide the story

01. Why aren't eigenvalues enough?

02. What does a kernel change?

03. How does KEDMD inherit geometry?

04. What does Kreiss measure?

05. How do pseudospectra help?

06. How should we choose kernels?

Theoretical illustrations and actual experimental evidence are explicitly labeled throughout.

Lesson 01 / Spectral stability

Can a stable matrix amplify a perturbation?

Yes. The spectral radius describes asymptotic behavior, but not necessarily the maximum norm at intermediate times.

\[B_\alpha=\begin{pmatrix}r&\alpha\\0&r\end{pmatrix},\qquad 0<r<1,\quad\sigma(B_\alpha)=\{r\}.\]

Keep the eigenvalues fixed. Change the off-diagonal coupling. Observe what happens.

Lesson 01 / Live computation · illustrative matrix

Same eigenvalues, different growth

Non-normal coupling 2.0

Fixed spectral radius: 0.8

Peak over t=0,…,40: —

Peak time: —

\[B_\alpha^t=\begin{pmatrix}r^t&t\alpha r^{t-1}\\0&r^t\end{pmatrix}.\]

Exact singular-value computation for the illustrated 2×2 matrix; not paper data.

Lesson 02 / Reproducing kernel geometry

What changes when we change the kernel?

\[\displaystyle x\longmapsto k_x:=k(\cdot,x)\in\mathcal H_k\]

A kernel determines inner products among sampled representers. Different kernels produce different geometric measurements of the same sampled states.

Representation-space amplification is not automatically physical-state amplification.

Lesson 02 / The four geometries studied

Four kernels, four induced geometries

Original kernel geometry comparison from research repository

Original scientific figure from the research repository. Kernel geometry affects the fitted representation and its induced norms.

Lesson 02 / Transport identity

Kernel-section transport

Given sampled states X and images Y=F(X), define the section transport on the sampled span.

\[T_{F,X}:\sum_i c_i k_{x_i}\mapsto\sum_i c_i k_{F(x_i)}.\]
\[\|T_{F,X}\|^2=\lambda_{\max}\!\left(G_X^{-1/2}G_YG_X^{-1/2}\right).\]

Finite sampled-span identity when G_X is positive definite. Singular Gram matrices require additional well-definedness conditions.

Lesson 03 / Geometry-normalized KEDMD

How does the Gram matrix enter the operator?

\[G_\lambda=G_X+\lambda I,\qquad A_\lambda=G_\lambda^{-1}A_X.\]
\[\widetilde A_\lambda=G_\lambda^{1/2}A_\lambda G_\lambda^{-1/2}.\]
\[\|A_\lambda^t\|_{G_\lambda}=\|\widetilde A_\lambda^t\|_2.\]

A similarity transform converts the Gram-induced norm into the Euclidean spectral norm. Across kernels, both the Gram geometry and fitted matrix may change.

Lesson 03 / The paper's central inequality

A geometry-sensitive bound

\[\chi_{k,\lambda}=\left\|G_\lambda^{-1/2}G_YG_\lambda^{-1/2}\right\|_2^{1/2}\]
\[\|\widetilde A_\lambda^t\|_2\leq \chi_{k,\lambda}^{\,t}.\]

This upper bound is not an assertion that the observed peak equals \(\chi_{k,\lambda}^{t}\).

The actual transient-growth curve depends on powers of the fitted normalized operator.

Lesson 03 / Research evidence

Measured finite-time amplification

Original linear E-I physical and representation transient growth figure

Original Linear E–I transient-amplification figure. Other systems are available in the main website laboratory.

Lesson 04 / Kreiss constant

The resolvent looks beyond eigenvalues

\[\mathcal K(B)=\sup_{|z|>1}(|z|-1)\|(zI-B)^{-1}\|_2.\]
\[\mathcal K(B)\leq \sup_{t\geq0}\|B^t\|_2.\]

When the spectrum lies in the closed unit disk, the finite-dimensional Kreiss matrix theorem also provides a dimension-dependent upper bound on the power supremum.

For matrices with spectral radius slightly above one, the infinite-horizon power supremum can be unbounded; don't apply the stable-matrix upper bound without checking hypotheses.

Lesson 04 / Numerical estimate vs. exact supremum

How is the Kreiss constant estimated?

\[\widehat{\mathcal K}_{\mathcal G}(B)=\max_{z\in\mathcal G,\,|z|>1}\frac{|z|-1}{\sigma_{\min}(zI-B)}.\]
31²

Coarse grid

61²

Reported estimate

121²

Refinement

Finite-grid maximization is a lower estimate of the exact supremum over the full exterior domain. The paper's convergence checks are empirical, not a certified global error bound.

Lesson 04 / Original numerical evidence

Kreiss estimate: kernel comparison

61×61 discrete Kreiss estimates, extracted from the original CSVs.

Measured finite-horizon peak growth

These are distinct diagnostics; matching kernel orderings do not imply equal magnitudes.

Lesson 04 / Grid refinement

Does the resolvent estimate converge?

Kreiss estimate convergence across grid resolutions

Original grid-convergence figure. The system selector on the previous slide also changes this figure.

Lesson 05 / Pseudospectra

Where is the resolvent large?

\[\sigma_\varepsilon(B)=\left\{z:\sigma_{\min}(zI-B)<\varepsilon\right\}\]

For a non-normal matrix, pseudospectral contours can extend far beyond the eigenvalue locations.

\[\mathcal K(B)\geq\frac{r_\varepsilon(B)-1}{\varepsilon},\quad r_\varepsilon(B)>1.\]

The bound follows from the resolvent definition and pseudospectral radius, with the usual convention for ε-pseudospectra.

Lesson 05 / Original pseudospectral contours

What the contours reveal

Original KEDMD pseudospectral contours

Original paper figure; select another system in Lesson 04 to update this plot.

Lesson 06 / Kernel choice

Is the kernel with the largest peak always best?

No universal answer follows from transient growth alone. Kernel choice also changes conditioning, regularization sensitivity, and the target of the analysis.

\[\displaystyle k_{\mathrm{preferred}}\;\neq\;\underset{k}{\operatorname{arg\,max}}\;\mathcal G_k\quad\text{in general}.\]

Here, \(\mathcal G_k\) denotes the measured peak representation-space amplification. A Pareto view makes competing diagnostics explicit without asserting a universal kernel ranking.

Lesson 06 / Pareto evidence

Dominance and trade-offs

Original kernel Pareto dominance figure

Original Pareto/dominance figure from the paper's repository. Select a system in Lesson 04 to update this plot.

Course conclusion

What should you take away?

1. Stable eigenvalues need not exclude transient growth.

2. Kernels define the geometry in which amplification is measured.

3. Resolvent and pseudospectral diagnostics illuminate non-normality.

4. Empirical kernel rankings are system- and criterion-dependent.

Explore theorems ↗   ·   Kreiss laboratory ↗   ·   Pareto analysis ↗

Further reading

Continue with the original research

Accepted paper ↗

Source code and original data ↗

Return to interactive research website ↗

Scientific plots and numerical data are taken from the research repository. The 2×2 demonstration is an independently computed teaching example.