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Learning Koopman Eigenfunctions and Invariant Subspaces from Data: Symmetric Subspace Decomposition

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Pith's one-line read Applying EDMD forward and backward in time identifies exactly which dictionary functions are Koopman eigenfunctions and yields the maximal invariant subspace.

desk verdict Solid, novel Koopman results with a real but easily fixed bug in the streaming pseudocode. read the letter →

arxiv 1909.01419 v4 pith:LWBJRK5M submitted 2019-09-03 eess.SY cs.SYmath.DS

Masih Haseli , Jorge Cortés This is my paper · ORCID
classification eess.SYcs.SYmath.DS MSC 37C3037M1037N35
keywords Koopmanoperatoreigenfunctionsinvariantsubspacesextendeddynamicmodedecompositionsymmetricsubspacestreamingalgorithmdata-drivendynamicalsystemstotalleastsquares
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the functions in a given dictionary which evolve linearly under a dynamical system can be identified exactly by running Extended Dynamic Mode Decomposition (EDMD) both forward and backward in time. A function is shown to evolve linearly on the available data precisely when its coefficient vector is an eigenvector of the forward EDMD matrix and, with reciprocal eigenvalue, of the backward EDMD matrix. Under dense sampling of a compact state space, this condition characterizes genuine Koopman eigenfunctions almost surely. Building on this, the paper's Symmetric Subspace Decomposition (SSD) algorithm iteratively prunes the dictionary so that its output spans the maximal Koopman-invariant subspace in the span of the original dictionary, with a streaming variant that uses fixed memory and an approximate variant for dictionaries lacking informative eigenfunctions.

What carries the argument

The load-bearing object is the pair of EDMD matrices $K_f$ and $K_b$, together with the iterated null-space reduction used by the Symmetric Subspace Decomposition (SSD) algorithm. At each SSD iteration the algorithm forms the concatenated matrix $[A_i,B_i]$ of current forward and backward dictionary snapshots, computes a basis for its null space, and uses it to replace the dictionary by a smaller one, continuing until the two ranges coincide: $R(D(X)C)=R(D(Y)C)$ for the final full-column-rank matrix $C$. This range equality is exactly the finite-data certificate that the reduced dictionary spans a Koopman-invariant subspace, and the eigenvectors of $K_{\mathrm{SSD}}=(D(X)C)^\dagger(D(Y)C)$ are provably the linear evolutions, hence the Koopman eigenfunctions, in the span of the original dictionary.

What would settle it

On a system with a dictionary that is not Koopman-invariant, run SSD on a large dense sample, take each identified linear evolution $f(x)=D(x)v$ with eigenvalue $\lambda$, and compute $\sup_{x\in M}|f(Tx)-\lambda f(x)|$ on a fine grid; if the supremum does not converge to zero as the sample grows, the paper's almost-sure eigenfunction guarantee fails.

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Extended reading notes

Core claim

The central discovery is a necessary and sufficient, data-only test for linear evolution: for a dictionary $D$ and snapshot matrices $X,Y$ with $y_i=T(x_i)$, a function $f(x)=D(x)v$ satisfies $f(y_i)=\lambda f(x_i)$ for all observed $i$ if and only if $K_f v=\lambda v$ and $K_b v=\lambda^{-1}v$, where $K_f=D(X)^\dagger D(Y)$ and $K_b=D(Y)^\dagger D(X)$ are the forward and backward EDMD matrices. If the sampling is dense, this condition is also sufficient for $f$ to be a true Koopman eigenfunction with probability one. The paper then shows that the SSD algorithm, which repeatedly intersects the column ranges of the forward and backward dictionary snapshots via null-space computations, provably recovers the maximal Koopman-invariant subspace in the span of the dictionary and all Koopman eigenfunctions in that span.

Load-bearing premise

The asymptotic guarantees rely on the data being dense enough that a continuous function which vanishes at every sampled point must vanish everywhere on the state space.

Editorial extensions

If this is right

  • EDMD restricted to the SSD subspace has zero residual, so long-term prediction on that subspace is exact rather than merely approximate.
  • Every Koopman eigenfunction lying in the span of the original dictionary is recovered as an eigenvector of $K_{\mathrm{SSD}}$, without needing multi-step trajectory data.
  • The forward-backward condition gives a computable certificate for linear evolution on the observed data that is not fooled by functions that happen to fit EDMD in only one time direction.
  • The streaming version SSSD provably computes the same invariant subspace as batch SSD while storing only a fixed-size window of data.
  • When the dictionary lacks enough informative eigenfunctions, the Approximated-SSD variant produces a total-least-squares linear model whose in-sample error is controlled by a tunable tolerance parameter.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same forward-backward eigenvector symmetry could serve as a regularizer for dictionary-learning and neural-network Koopman methods, which currently optimize mostly one-step forward prediction error.
  • The range-intersection pruning is a general linear-algebraic primitive that may transfer to stochastic or transfer-operator settings, where forward and adjoint/generator action play the role of the backward pass.
  • In finite-data practice, the almost-sure guarantee motivates a hold-out validation procedure: compute the continuous residual $f(Tx)-\lambda f(x)$ on a fine grid and monitor whether its maximum shrinks as the sample grows.
  • The monotonicity of SSD under data addition suggests an active data-acquisition scheme that adds snapshots where the current invariant dictionary's residual is largest, potentially shrinking the invariant subspace faster than random sampling.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper considers data-driven identification of Koopman eigenfunctions and Koopman-invariant subspaces from snapshot pairs (X,Y) of a discrete-time dynamical system. It first gives a forward-backward EDMD condition (Theorem 4.3) that is necessary and sufficient for a dictionary function to evolve linearly with nonzero eigenvalue on the observed data. It then introduces the Symmetric Subspace Decomposition (SSD) algorithm, proves that its output is the maximal subspace satisfying the corresponding range condition (Theorem 5.1), and, under a dense-sampling assumption, that the SSD limit is the maximal Koopman-invariant subspace in the dictionary span and that its eigenvectors are Koopman eigenfunctions (Theorems 5.7 and 5.8). A streaming version (SSSD), an approximation version (Approximated-SSD), and two numerical examples are also presented.

Significance. If the stated results hold, the paper is a useful contribution to the data-driven Koopman literature: Theorem 4.3 gives a clean finite-data characterization of linear evolution, and the maximality result in Theorem 5.8 is stronger than what is typically proved for EDMD-based methods. The linear-algebraic core, especially the proofs of Theorems 4.3, 5.1, and 6.3, is carefully reasoned and appears correct. The main reservations are that the streaming algorithm is not reproducible as printed and that the passage from Assumption 4.4 to almost-sure density is asserted rather than proved. These are fixable, but they are load-bearing for the corresponding claims.

major comments (2)
  1. [Section VI, Algorithm 2 (lines 21–24)] The pseudocode for SSSD contains a control-flow error: line 21 executes `return Ci` inside the `while 1` loop, before the `i←i+1` update at line 22 and before the new-data updates at lines 23–24. As printed, the algorithm terminates after processing the single snapshot pair (x_{S+1}, y_{S+1}) and never refines the subspace as additional data arrive. Consequently, Theorem 6.3, which equates the i-th SSSD output with SSD on all data up to S+i, does not apply to the printed procedure, and the reported 96% speedup in Example 8.1 is not reproducible from Algorithm 2 as written. The fix is local (move the output outside the loop, or change the return semantics to an emit-and-continue convention), but it is necessary for the streaming contribution.
  2. [Assumption 4.4 and Theorem 4.6(b)] The proofs of Theorems 4.6(b), 5.7(b), and 5.8 assert that S∞ = ⋃N S_N is dense in M almost surely merely from p_N → 1. This step is not immediate and should be justified. The conclusion is in fact correct for the nested sets S_N: writing A_N for the event that S_N is α(1/N)-dense, Fatou's lemma gives P(limsup_N A_N) ≥ limsup_N P(A_N) = 1, and on limsup A_N the union S∞ is dense. However, the paper should state this argument explicitly and should make clear that the events A_N are defined on a common probability space. As written, the proof contains only the assertion, and this is a load-bearing point for the asymptotic guarantees.
minor comments (4)
  1. [Section III and Theorems 4.3, 5.5] The problem statement says the goal is to find all Koopman eigenfunctions in span(D), but Theorems 4.3 and 5.5 restrict to λ ≠ 0. Note also that, under Assumption 3.1, a zero-eigenvalue eigenfunction cannot be represented nontrivially in span(D), since D(Y)v = 0 would contradict the full column rank of D(Y). Please add a sentence clarifying this qualification.
  2. [Theorem 5.5 proof] In the complex-vector case, the proof states 'there exists a real vector z such that E = CSSDz'. Since E is a matrix with two columns, the object should be a matrix Z (or the argument should be rewritten column by column); the subsequent multiplication by r = [1, j]^T then produces the vector w = Zr. This is likely a typo, but it makes the proof hard to read.
  3. [Theorem 4.6(b) proof] The line 'f(yi) = λ f(xi) v' contains a spurious trailing v; it should read f(yi) = λ f(xi).
  4. [Algorithm 3, Step 8] The notation V^T_{kmin:m} is ambiguous; the text should specify whether it denotes rows kmin through m of V^T or columns kmin through m of V. The subsequent discussion indicates the latter, but the notation should be made precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain is self-contained, and the self-citations are peripheral pointers rather than load-bearing assumptions.

full rationale

I walked the derivation chain from the EDMD definition through the forward–backward characterization, the SSD algorithm, and the streaming extension. Lemma 4.1 and Theorem 4.3 are genuine algebraic consequences of the definitions of Kf = D(X)^†D(Y) and Kb = D(Y)^†D(X), not restatements of those definitions: Theorem 4.3 uses Assumption 3.1 and a norm/orthogonal-decomposition argument to prove that reciprocal eigenvector conditions imply linear evolution on the data. Theorem 4.6 upgrades this to true Koopman eigenfunctions by defining the continuous residual h = f∘T − λf, using the data condition to show h vanishes on the sampled set, and invoking Assumption 4.4 to conclude h vanishes on the compact state space almost surely. This is a substantive continuity/density argument, not a renaming of an input. The SSD output is, by construction, a maximal subspace satisfying R(D(X)C) = R(D(Y)C), but Theorem 5.8 does not stop there: it proves that the asymptotic intersection of the SSD subspaces is actually Koopman-invariant and contains every Koopman-invariant subspace in span(D), using the same dense-sampling argument. That is a nontrivial bridge from the data-level range equality to the operator-level invariance. Theorem 6.3 likewise proves the SSD–SSSD equivalence by induction using maximality and monotonicity rather than assuming it. The self-citations, [1] and [37], are not load-bearing: [1] is cited as a preliminary version and for a counterexample, while [37] is an unrelated noise-resilient EDMD extension; neither supplies a premise needed for the main theorems. The paper also explicitly acknowledges the limitation that Approximated-SSD gives no out-of-sample bound, which is a scope statement, not a circular step. The skeptical observation about Algorithm 2's return inside the while loop is a reproducibility defect in the pseudocode, not an instance of circular reasoning. Overall, no prediction or first-principles result reduces by construction to a fitted parameter, a self-citation, or an input definition.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper's guarantees rest on full column rank of dictionary data matrices (Assumption 3.1), dense sampling (Assumption 4.4), continuity of the dynamics and dictionary, and Assumption 6.1 for the streaming variant. No parameters are fitted to data; the design parameter epsilon tunes the approximate variant.

free parameters (2)
  • epsilon (accuracy parameter) = 10^-12 in Example 8.1, 10^-3 in Example 8.2
    Chosen by hand to truncate small singular values in the nullspace approximation; controls the trade-off between subspace dimension and approximation accuracy. Not fitted to a target result.
  • Signature size S in SSSD = S = 10 in Example 8.1
    User-selected number of signature snapshots required by Assumption 6.1. Any S that yields full-rank dictionary matrices is in principle valid.
assumptions (5)
  • domain assumption Assumption 3.1: D(X) and D(Y) have full column rank
    Used in Lemma 4.1, Theorem 4.3, and throughout; guarantees dictionaries are linearly independent on data and pseudo-inverse identities hold.
  • domain assumption Assumption 4.4: almost sure dense sampling from a compact state space
    Used to promote data-linear evolutions to true Koopman eigenfunctions and to establish maximality of the SSD output in the limit.
  • domain assumption Assumption 6.1: full rank signature dictionary matrices
    Required by SSSD to guarantee the algorithm is well-defined and equivalent to SSD.
  • domain assumption Continuity of T and dictionary functions
    Needed for the density argument that a continuous residual vanishing on a dense set vanishes everywhere (Theorems 4.6, 5.8).
  • standard math Standard linear algebra results (SVD, Eckart-Young theorem, range-nullspace lemmas)
    Used in proofs of low-rank approximation and total least squares solution in Section VII and the Appendix.

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Pith. "Pith review of Learning Koopman Eigenfunctions and Invariant Subspaces from Data: Symmetric Subspace Decomposition." pith.science (2026). https://pith.science/paper/LWBJRK5M

@misc{pith2026190901419,
  author       = {Pith},
  title        = {Pith review of: Learning Koopman Eigenfunctions and Invariant Subspaces from Data: Symmetric Subspace Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LWBJRK5M}},
  note         = {Machine review of arXiv:1909.01419}
}
read the original abstract

This paper develops data-driven methods to identify eigenfunctions of the Koopman operator associated to a dynamical system and subspaces that are invariant under the operator. We build on Extended Dynamic Mode Decomposition (EDMD), a data-driven method that finds a finite-dimensional approximation of the Koopman operator on the span of a predefined dictionary of functions. We propose a necessary and sufficient condition to identify Koopman eigenfunctions based on the application of EDMD forward and backward in time. Moreover, we propose the Symmetric Subspace Decomposition (SSD) algorithm, an iterative method which provably identifies the maximal Koopman-invariant subspace and the Koopman eigenfunctions in the span of the dictionary. We also introduce the Streaming Symmetric Subspace Decomposition (SSSD) algorithm, an online extension of SSD that only requires a small, fixed memory and incorporates new data as is received. Finally, we propose an extension of SSD that approximates Koopman eigenfunctions and invariant subspaces when the dictionary does not contain sufficient informative eigenfunctions.

Figures

Figures reproduced from arXiv: 1909.01419 by the authors.

Figure 1
Figure 1. Relative (left) and angle (right) prediction errors on the original and SSD subspaces for system (59) on a trajectory of length M = 20. Example 8.2: (Duffing Equation): Here, we investigate the efficacy of the proposed methods in approximating Koopman eigenfunctions and invariant subspaces. Consider the Duffing equation [35, Section 4.2] x˙ 1 = x2 x˙ 2 = −0.5x2 + x1 − x 3 1 , (61) with state x T = [x1, x2] T . The s… view at source ↗
Figure 2
Figure 2. (left) predicts that the trajectories eventually converge to one of the stable equilibria [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Relative (left) and angle (right) prediction errors on Approximated-SSD and original subspaces for system (61) on a trajectory of length M = 30. IX. CONCLUSIONS We have studied the characterization of Koopman-invariant subspaces and Koopman eigenfunctions associated to a dy￾namical system by means of data-driven methods. We have shown that the application of EDMD over a given dictionary forward and backward in time … view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.