Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data

Baike She; George Chiu; Lei Xin; Qi Dou; Shreyas Sundaram

REVIEW 1 major objections 4 minor 35 references

Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that restart experiments with deterministic initial perturbations can consistently recover the linear part of a nonlinear system, at a worst-case rate of $O(N^{-1/4})$ when the feasible region contains the origin.

desk verdict Useful and well-scoped paper on learning linearized models under initialization constraints, but Theorem 1's printed noise bound has a 1/q scaling slip that makes the advertised O(N^{-1/4}) consistency false as written; the fix is straightforward and the underlying idea is sound. read the letter →

arxiv 2505.04954 v1 pith:KYKRZXAB submitted 2025-05-08 stat.ML cs.LGcs.SYeess.SY

Lei Xin , Baike She , Qi Dou , George Chiu , Shreyas Sundaram This is my paper · ORCID

classification stat.MLcs.LGcs.SYeess.SY

keywords systemidentificationlinearizedmodelnonlineardynamicsfinite-sampleguaranteesmultipletrajectoriesregularizedleastsquaresinitializationconstraintspersistentexcitation

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 asks a practical question: when a physical system is genuinely nonlinear but a control engineer only needs its linearized model, how much data and what kind of experiments are needed to recover that model with a guarantee? It proposes collecting $N$ one-step trajectories from restart experiments whose initial state-plus-input points are chosen deterministically as small perturbations $\pm q e_i$ around a center $m$ inside a feasible region, then fitting $\Theta = [A\;B]$ by regularized least squares. The central result is a finite-sample bound on $\|\hat{\Theta} - \Theta\|$ that splits into noise, nonlinearity, and regularization terms and exhibits a bias-variance-like trade-off in the perturbation size $q$. When the feasible region contains the origin and $q$ is chosen to shrink as $N^{-1/4}$, the bound gives worst-case consistency at rate $O(N^{-1/4})$; for a truly linear system it recovers the familiar $O(N^{-1/2})$ rate. This matters because existing finite-sample identification guarantees mostly assume the dynamics are exactly linear or require a single long trajectory with random inputs and globally Lipschitz nonlinearities, both of which can fail in mild nonlinear settings.

What carries the argument

The load-bearing object is the Taylor decomposition of the nonlinear map around the origin, $x_{k+1} = \Theta z_k + w_k + r_k$ with $\Theta = [A\;B]$ and remainder $r_k = h(z_k)$ whose components are bounded by $\beta\|z_k\|_1^2$ (Assumption 1). Algorithm 1's sign-alternating deterministic perturbations $\pm q e_i$ around $m$ are the second piece: they make the design covariance $ZZ'$ well conditioned, with $\lambda_{\min}(ZZ') \ge Nq^2/[2(n+p)]$ (Lemma 1), while the sign changes cancel the center $m$ in cross terms; this is what lets small $q$ keep the nonlinearity bias low without losing persistent excitation. Algorithm 2's regularized least squares with ridge parameter $\lambda$ produces the closed-form estimator, and the proof decomposes the error via the triangle inequality into a bias term from $\lambda$, a nonlinearity term from $RZ'(ZZ'+\lambda I)^{-1}$, and a noise term bounded through a sub-Gaussian self-normalized martingale inequality. The regularization parameter is not merely numerical: it damps the noise and nonlinearity terms at the cost of an irreducible $\|\Theta\|$ bias.

What would settle it

Run the scalar system $x_{k+1} = x_k + x_k^2 + w_k$ with $\Theta = 1$, small sub-Gaussian noise variance, $m=0$, $\lambda=0$, and $q = c_0 N^{-1/4}$. If the empirical $\|\hat{\Theta} - 1\|$ across many independent trials fails to shrink at the predicted $O(N^{-1/4})$ rate as $N$ grows, or fails to shrink to zero in the noiseless case, then the claimed consistency rate and bound are disproved.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1: under Assumption 2, meaning that the deterministic points $m \pm q e_i$ lie in the feasible region $S$, and under Assumption 1, meaning that the Taylor remainder $r_k$ satisfies $|r_{i,k}| \le \beta \|z_k\|_1^2$ near the origin, the regularized least-squares estimate $\hat{\Theta} = XZ'(ZZ' + \lambda I)^{-1}$ produced by Algorithms 1 and 2 satisfies, with probability at least $1-\delta$, a three-term error bound. The noise term scales roughly like $\sigma_w/(q\sqrt{N})$, the nonlinearity term like $\beta q$, and the regularization term like $\lambda\|\Theta\|$ divided by the covariance size. The author's conclusion is that consistent learning of the linearized dynamics is possible with finitely many short trajectories even when the nonlinearity is not globally Lipschitz: when $S$ is open and contains $0$, taking $m=0$, $\lambda=0$, and $q = c_0 N^{-1/4}$ yields $\|\hat{\Theta} - \Theta\| = O(N^{-1/4})$. The same framework yields small error for convex feasible regions that do not contain the origin, provided the center $m$ is close enough to $0$ and $N$ is large, and for perfectly linear systems the bound reduces to an $O(1/\sqrt{N})$ rate comparable with existing random-input results.

Load-bearing premise

The guarantee rests on resetting each experiment's initial state and input exactly to the deterministic points $m \pm q e_i$; if initializations are only approximate, the proof's covariance lower bound and nonlinearity remainder bound no longer follow.

Editorial extensions

If this is right

Engineers can obtain a certified linear model for controller design using many one-step restart experiments, and the guarantee holds regardless of the spectral radius of $A$, so even unstable nonlinear systems are in scope.
When initialization can reach a neighborhood of the reference point, the perturbation size $q$ should be chosen to balance noise and nonlinearity; a default of $q \propto N^{-1/4}$ gives guaranteed worst-case consistency at rate $O(N^{-1/4})$.
In the special case of perfectly linear dynamics, the bound reproduces the $O(1/\sqrt{N})$ sample complexity of established multiple- and single-trajectory linear identification methods, so the approach does not sacrifice the linear benchmark.
If the feasible region excludes the origin, the center $m$ should be as close to the origin as constraints allow, and a larger regularization $\lambda$ becomes useful when noise, nonlinearity, or distance from the origin is large.
State predictions from the learned model are close to the true nonlinear map near the origin, with prediction error dominated by the identification error $\|\hat{\Theta} - \Theta\|$ rather than by the nonlinear terms when $\|z\|$ is small (Proposition 1).

Reading between the lines

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

Beyond the paper: the same alternating-perturbation design should extend to estimating the quadratic term of the Taylor expansion, since finite-difference combinations of the one-step outputs isolate second-order coefficients; this would give a direct data-driven check of Assumption 1's constant $\beta$.
Beyond the paper: a fully adaptive experiment could choose $q$ per batch by estimating the noise variance and nonlinearity curvature from earlier batches, potentially achieving faster than worst-case convergence; Theorem 1's trade-off suggests this is feasible but does not analyze it.
Beyond the paper: with inexact initialization modeled as an extra bounded disturbance, the two proofs that fail (Lemma 1 and Lemma 3) could be patched by adding an initialization-error term to the bound; the numerical robustness shown in Section 5.1 is evidence this patch would work, but the paper leaves the proof open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Quick take: the paper attacks a real gap—finite-sample guarantees for identifying the Jacobian of a noisy nonlinear system when you can only probe from a feasible region. The deterministic multi-trajectory design, the ridge regression analysis, and the explicit trade-off between bias from nonlinearity and variance from noise are genuinely new compared with the authors' prior CDC paper, and the numerics back the qualitative storyline. The writing is clear and the proof structure is honest.

The problem: the printed Theorem 1 is internally inconsistent. The 'error due to noise' term carries a factor 1/q in the numerator in addition to the 1/q that comes from the singular-value lower bound. Plugging the proof's own equations (9)-(10) and Lemma 2 into the decomposition gives a noise term of order (1/sqrt(N q^2)) sqrt(log), i.e., no extra 1/q. As printed, with q ~ N^{-1/4} the noise term is O(1), so the consistency claim does not follow from the statement. The corrected scaling, 1/(q sqrt N), balances the nonlinearity term O(beta q) at q ~ N^{-1/4}, giving the advertised rate. So this is very likely a typographical slip, but it is load-bearing: a referee has to be able to verify the bound as stated, and right now the theorem as written is false.

Two secondary points. First, the theory assumes perfect resets to deterministic initial points z0 = m ± q e_i. Lemma 1 and Lemma 3 depend on that exactly. The robustness experiment in Section 5.1 only adds small Gaussian perturbations and shows the error stays small; that's a legitimate scope limitation, clearly flagged by the authors, not a hidden flaw. Second, the numerical comparison with a single trajectory under i.i.d. Gaussian inputs convincingly demonstrates the failure mode the paper is motivated by.

The citation pattern is fine: self-citations to the conference version and to [33] for matrix concentration lemmas are standard and don't carry the result's weight. I found no circular parameter fitting.

Recommendation: send it to review. A competent referee can check the corrected bound in an afternoon. The problem is well-motivated, the algorithm is practical, and the corrected result is likely to stand. My verdict is conditional: fix the statement of Theorem 1, double-check Lemma 3's algebra, and either extend the initialization-robustness analysis or state the exact-reset scope more prominently.

Referee Report

1 major / 4 minor

Summary. The paper studies finite-sample identification of the linearized dynamics Theta = [A B] of a discrete-time nonlinear system x_{k+1} = f(x_k,u_k) + w_k around the origin, under the constraint that experiments can only be initialized inside a feasible set S. It proposes a deterministic multi-trajectory data acquisition scheme (Algorithm 1) that cyclically initializes the state-input vector at m +/- q e_i within S, followed by a regularized least-squares estimator (Algorithm 2). The main result (Theorem 1) bounds ||hat Theta - Theta|| by three terms: a noise term, a nonlinearity term, and a regularization term. The paper claims that when S is an open set containing the origin, choosing q proportional to N^{-1/4} gives consistency at rate O(N^{-1/4}), and that when S does not contain the origin a small error is still possible if S is close enough to the origin. Numerical experiments on a pendulum-like system and on a strongly nonlinear system illustrate the trade-offs and show better behavior than single-trajectory Gaussian-input identification. The paper explicitly notes that exact resetting is assumed and that initialization perturbations are only tested numerically.

Significance. If the main theorem is corrected, the paper makes a useful contribution: it provides one of the first finite-sample guarantees for learning a local linear model of a nonlinear stochastic system with a constrained initialization region, without requiring global Lipschitzness of the unmodeled dynamics. The decomposition of the error into noise, nonlinearity, and regularization terms is clean, the data acquisition scheme is simple and deterministic, and the numerical experiments support the qualitative predictions. The proofs are self-contained in the appendix, including persistent excitation, sub-Gaussian concentration, and Taylor remainder bounds. The exact-initialization limitation is disclosed honestly, and robustness to initialization jitter is offered only as empirical evidence. The claimed O(N^{-1/4}) rate, if supported, would be a useful improvement over the existing practice for nonlinear system identification.

major comments (1)

[Theorem 1, Eq. (8)] The displayed "Error due to noise" term contains a spurious factor 1/q and an inconsistent denominator. In the proof, Eqs. (9)-(10) combined with Lemma 2 and Lemma 1 give the noise contribution bounded by 3 sigma_w sqrt(log(9n/delta) + (n+p) log(1 + ...)) / sqrt(N q^2/(2(n+p)) + lambda), with no 1/q factor. For m = 0 and lambda = 0, the printed term scales as 1/(q^2 sqrt(N)), so the recommended choice q proportional to N^{-1/4} leaves this term O(1), contradicting the claimed O(N^{-1/4}) consistency in Section 4.2. The proof's own bound balances the O(beta q) nonlinearity term exactly at q proportional to N^{-1/4}, so Eq. (8) and the related rate statements in Section 4.2 must be corrected. I recommend replacing the noise term by 3 sigma_w sqrt(L) / sqrt(N q^2/(2(n+p)) + lambda) or an equivalent conservative upper bound without the 1/q factor, and re-deriving the displayed constants in the theorem.

minor comments (4)

[Section 7.4, proof of Lemma 3] The square-root expression following Eq. (24) is ambiguous as typeset; it should read sqrt(2(n+p)/(N q^2 + 2 lambda (n+p))) + 2 sqrt(lambda)(n+p)/(N q^2 + 2 lambda (n+p)). The current typesetting suggests that the second summand is inside the radical, which is dimensionally inconsistent.
[Theorem 1, Eq. (8) and Lemma 2] The logarithmic factor in Theorem 1 omits the term zeta = 4 lambda (n+p)/N that appears in the denominator of Lemma 2's log argument. This omission is a valid loosening of the bound, but it should be explicitly acknowledged so that the two displays are visibly consistent.
[Section 5.1] The numerical robustness experiment adds Gaussian perturbations to the designed initial conditions, but the formal analysis assumes exact resetting. The paper's own caveat at the end of Section 5.1 correctly states that a detailed analysis is left for future work; I recommend making this limitation explicit in the contributions or abstract as well, since the formal guarantee does not cover inexact initialization.
[Theorem 1, Eq. (8)] The expression sqrt(lambda N n beta^2 b^2 q^4) in the regularization term should be parenthesized or written as sqrt(lambda N n) beta b q^2 to avoid ambiguity about what is inside the radical.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-sample bound follows from Taylor's theorem, deterministic persistent excitation, and generic concentration/matrix lemmas, with no fitted parameter renamed as a prediction.

full rationale

The paper's target Θ is defined by the Taylor expansion in Eq. (2), not by the data; Assumption 1 supplies an independent uniform quadratic bound on the remainder. The estimation error decomposition in Eq. (6) is exact algebra from the batch relation X = ΘZ + W + R. Theorem 1 then assembles Lemma 1 (persistent excitation from Algorithm 1's deterministic ±q pattern), Lemma 2 (sub-Gaussian concentration), and Lemma 3 (remainder bound) with explicit constants. The only self-cited ingredients are Lemma 4 and Lemma 6 from [33], which are generic concentration and positive-semidefinite comparison statements whose assumptions do not include the target identification rate, so they constitute independent support rather than a self-referential premise. The paper explicitly acknowledges the exact-initialization limitation in Section 5.1 and leaves initialization-noise analysis to future work; that is a reported limitation, not a circular step. The apparent 1/q factor discrepancy between the displayed noise term of Eq. (8) and the proof's Lemma 2 bound is a correctness/scaling issue, not a case of the conclusion being assumed as an input, and is therefore outside this circularity pass.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard smoothness and noise assumptions (Assumptions 1 and 2), the ability to reset initial conditions exactly, and a set of generic concentration and matrix inequalities. The user-chosen parameters q, m, and lambda are design choices that trade off bias and variance; none are fitted to data. No new physical entities are introduced.

free parameters (3)

q
Norm constraint parameter in Algorithm 1; balances bias from nonlinearity (small q) against noise error (large q). Chosen by the user, not fitted to data.
m
Center point of the probing set; chosen as close to the origin as the feasible region S allows to reduce the nonlinearity error term. A user choice.
lambda
Ridge regularization parameter in Algorithm 2; larger lambda reduces noise and nonlinearity terms at the cost of an increasing bias term ||Theta||. Selected by cross-validation in experiments, not fitted to derive the theory.

assumptions (5)

domain assumption Each component function of f has continuous second-order partial derivatives, so the Taylor remainder satisfies |r_{i,k}| <= beta ||z_k||_1^2 locally (Assumption 1).
This is the key regularity condition that replaces the global Lipschitz assumption in [27]; it is mild for typical smooth systems but is assumed, not proven.
domain assumption The feasible region S contains the points m +/- q e_i for all i (Assumption 2).
Ensures Algorithm 1 is implementable and yields the persistent excitation lower bound in Lemma 1.
domain assumption f(0)=0 and the state and input are perfectly measured.
The linearization is taken around the origin; measurement noise is not modeled.
domain assumption Process noise w_k are independent sub-Gaussian with parameter sigma_w^2.
Needed for the concentration bound in Lemma 2.
standard math Sub-Gaussian concentration inequality (Lemma 4, from the authors' earlier paper [33]) and matrix inequalities (Lemmas 5,6) are correct.
These are cited results used in the proofs; [33] is self-cited but the lemmas are generic.

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Cite this review

Pith. "Pith review of Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data." pith.science (2026). https://pith.science/paper/KYKRZXAB

@misc{pith2026250504954,
  author       = {Pith},
  title        = {Pith review of: Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYKRZXAB}},
  note         = {Machine review of arXiv:2505.04954}
}

read the original abstract

The identification of a linear system model from data has wide applications in control theory. The existing work that provides finite sample guarantees for linear system identification typically uses data from a single long system trajectory under i.i.d. random inputs, and assumes that the underlying dynamics is truly linear. In contrast, we consider the problem of identifying a linearized model when the true underlying dynamics is nonlinear, given that there is a certain constraint on the region where one can initialize the experiments. We provide a multiple trajectories-based deterministic data acquisition algorithm followed by a regularized least squares algorithm, and provide a finite sample error bound on the learned linearized dynamics. Our error bound shows that one can consistently learn the linearized dynamics, and demonstrates a trade-off between the error due to nonlinearity and the error due to noise. We validate our results through numerical experiments, where we also show the potential insufficiency of linear system identification using a single trajectory with i.i.d. random inputs, when nonlinearity does exist.

Figures

Figures reproduced from arXiv: 2505.04954 by the authors.

**Figure 2.** System identification error using a single tra [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** System identification error under initialization [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: System identification error using different reg [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [33]

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Lei Xin, Lintao Ye, George Chiu, and Shreyas Sundaram. Learning dynamical systems by leveraging data from similar systems. arXiv preprint arXiv:2302.04344 , 2023

arXiv 2023

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work page 2021

[3] [2]

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Masanao Aoki. State space modeling of time series. Springer Science & Business Media, 2013

work page 2013

[4] [3]

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Susan Athey. The impact of machine learning on economics. In The economics of artificial intelligence: An agenda , pages 507–547. University of Chicago Press, 2018

work page 2018

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NN Chan and Man Kam Kwong. Hermitian matrix inequalities and a conjecture. The American Mathematical Monthly, 92(8):533–541, 1985

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S ¸tefan Cobza¸ s, Radu Miculescu, Adriana Nicolae, et al. Lipschitz functions. Springer, 2019

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