Finite Sample Analysis of Subspace Identification for Stochastic Systems

Shuai Sun; Weikang Hu; Xu Wang

REVIEW 3 major objections 4 minor 50 references

Finite Sample Analysis of Subspace Identification for Stochastic Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Finite-sample subspace identification of stochastic LTI systems achieves $O(1/\sqrt{N})$ error for $A,C,K$, $O(N^{-1/(2n)})$ for poles, and needs super-polynomial samples in the state-to-output ratio for constant error.

desk verdict Finite-sample matrix and pole bounds for SIM are solid and worth a referee's time, but the advertised super-polynomial sample-complexity lower bound is not established and the key imported lemma has a mismatch with the paper's own assumptions. read the letter →

arxiv 2501.18853 v5 pith:HVNMKBOO submitted 2025-01-31 eess.SY cs.SY

Shuai Sun , Weikang Hu , Xu Wang This is my paper · ORCID

classification eess.SYcs.SY MSC 93E1293B30

keywords systemidentificationsubspacemethodfinitesampleanalysisstochasticLTIsystemsblockHankelmatrixcomplexitypolesKalmanfiltergain

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 tries to prove that the subspace identification method (SIM), applied to an $n$-dimensional discrete-time linear time-invariant system with no external inputs and $m$ outputs, has finite-sample error guarantees driven by the number $N$ of independent output trajectories. The claimed rates, up to logarithms, are $O(1/\sqrt{N})$ for the estimated system matrices $A$, $C$, and the Kalman gain $K$, and $O(N^{-1/(2n)})$ for the estimated system poles. The paper also claims that forcing the estimation error below a constant requires a sample size that grows super-polynomially in the state-to-output ratio $n/m$, because the smallest singular value of the associated block Hankel matrix decays super-polynomially. A reader should care because these are non-asymptotic, data-independent bounds for a widely used identification method in the stochastic, input-free setting.

What carries the argument

The load-bearing object is the block Hankel matrix $H_T=\Gamma_T K_T$ built from the extended observability matrix of $(A,C)$ and the reversed extended controllability matrix of $(A-KC,K)$ in the innovation form. Its smallest nonzero singular value $\sigma_n(H_T)$ controls the robustness of the whole SIM pipeline: the perturbation lemmas show that errors in $A$, $C$, $K$ are bounded by multiples of $\|\widehat H_T-H_T\|/\sqrt{\sigma_n(H_T)}$, while Lemma 2 upper-bounds $\sigma_n(H_T)$ by a quantity decaying super-polynomially in $n/m$. The second object is the regression identity $\widehat H_T=Y_f Y_p^\dagger$, whose error splits into a Gaussian cross term and a Kalman-filter truncation bias; Gaussian concentration bounds on $Y_pY_p^\top$ and on the cross terms supply the $1/\sqrt N$ rate once $T\sim\log N$.

What would settle it

Compute the smallest singular value $\sigma_n(H_T)$ for a non-normal but diagonalizable $A$ with real distinct eigenvalues in $[-1,1]$, for example $A=PDP^{-1}$ with $D=\mathrm{diag}(0.99,0.98,\dots)$ and a badly conditioned $P$, with $m=1$ and $T\approx\log N$, and compare it with the bound $\rho^{-(n-1)/(2m)}\log(2mT)$. If $\sigma_n(H_T)$ does not decay at that super-polynomial rate, Lemma 2 and the derived sample complexity fail; if it does decay, the paper's hardness conclusion survives for non-normal systems.

Watch

Extended reading notes

Core claim

The central claim is an end-to-end finite-sample analysis of SIM for a minimal stochastic LTI system in innovation form. The estimate $\widehat H_T$ of the block Hankel matrix $H_T=\Gamma_T K_T$ is formed by least-squares regression of future outputs on past outputs, then rank-$n$ truncated and factored to recover $\widehat\Gamma_T$ and $\widehat K_T$, and hence $\widehat A$, $\widehat C$, and $\widehat K$ up to a unitary similarity. The paper proves that with $T$ of order $\log N$ the regression error is $O((\log N)^{5/2}/\sqrt N)$, and that this error propagates to $A$, $C$, $K$ at the same $1/\sqrt N$ rate scaled by $n/\sigma_n(H_T)$, because $\sigma_n(H_T)$ controls how near-observable the weakest mode is. For the poles, the Hausdorff distance between true and estimated spectra is bounded by a power of the $A$-error, yielding the slower $N^{-1/(2n)}$ rate. The same $\sigma_n(H_T)$ bound implies that the sample size needed for constant error grows like a power of $\rho^{(n-1)/(2m)}$ with $\rho=e^{\pi^2/4}$, i.e., super-polynomially in $n/m$.

Load-bearing premise

The conclusion that constant-error identification needs super-polynomial sample size rests on a singular-value bound imported from another paper that assumes the system matrix is unitarily diagonalizable (normal), whereas the main theorem here only assumes real distinct eigenvalues; if that imported bound does not apply to non-normal $A$, the super-polynomial sample-complexity claim is unsupported.

Editorial extensions

If this is right

With $T$ chosen as $\Theta(\log N)$, the matrix identification errors $\|\widehat C-CU\|$, $\|\widehat K-U^\top K\|$, and $\|\widehat A-U^\top AU\|$ for a unitary $U$ fall at least as fast as $(\log N)^{O(1)}/\sqrt N$ with high probability.
The Hausdorff distance between the true poles and the estimated poles falls at least as $N^{-1/(2n)}$, meaning the pole estimates converge more slowly for larger state dimension.
For a constant-error target, the required number of trajectories must scale at least super-polynomially in $n/m$; high-dimensional state spaces with few outputs are therefore fundamentally harder for SIM-based identification.
The finite-sample guarantees are data-independent: they depend on $N$, $n$, $m$, $T$, noise covariances, and entry-size bounds for $C$ and $K$, not on the particular observed trajectories.
In the input-free setting, noise plays a double role: it excites the modes needed for identifiability, but it also inflates the least-squares cross term, a trade-off captured by the appearance of $\lambda_{\min}(R)$ in the denominator.

Reading between the lines

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

Inference: The super-polynomial sample complexity suggests that SIM-style identification is impractical for large $n/m$ in the no-input regime; a natural extension would be to test whether adding a known excitation input removes the $\sigma_n(H_T)$ penalty, since the paper's proof is restricted to the input-free case.
Inference: Because the pole bound passes through the Hausdorff distance via an $n$-th root, the $N^{-1/(2n)}$ rate is a worst-case over all $n$ poles; a finer per-pole analysis might show that only the weakly observable modes suffer the slow rate.
Inference: The same $\sigma_n(H_T)$ decay mechanism would likely appear in any identification method that reconstructs $A$ from a finite Hankel matrix of output covariances, suggesting the hardness is a property of the problem rather than of SIM specifically; a comparison against a least-squares innovation-form estimator would test this.
Inference: The numerical experiments cover only $n=4$; a direct check of the super-polynomial prediction would evaluate $\sigma_n(H_T)$ for non-normal $A$ with real distinct eigenvalues and increasing $n/m$.

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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

Short version: the matrix and pole bounds are real, the concentration argument is mostly coherent, and the paper deserves a serious referee. But the headline result about super-polynomial sample complexity in n/m is not proved, and the proof as written relies on an imported lemma whose assumptions don't match the paper's setting.

What's good: Theorem 1's analysis of the block Hankel regression error is careful. The three-step split — lower bound on the past-output covariance, cross-term bound, and Kalman truncation bias — is standard but executed cleanly, and the resulting O((log N)^{5/2}/sqrt(N)) rate with T ~ Theta(log N) is consistent with prior work. Lemma 1's perturbation bound is useful, and Theorem 3's pole bound is a legitimate corollary of the matrix bound plus eigenvalue perturbation theory. The simulations show the predicted matrix and pole trends, though they don't stress the n/m scaling.

The soft spots are where the reader's report finds them, and they are load-bearing for the advertised contribution. Equations (32)-(33) invert upper bounds and conclude that N 'must satisfy' a super-polynomial condition. That's logically invalid: an upper bound on the error gives a sufficient N to reach a target accuracy, not a necessary one. No lower bound on the estimation error of the SIM estimator or any estimator is proved. So the abstract's 'requires' should be downgraded to 'the upper bound has this shape' unless a real lower bound appears. That's still an interesting observation, but it is not the negative result the abstract claims.

The second issue is Lemma 2's use of Lemma 6. Lemma 6 assumes D is unitary diagonalizable, while Assumption 1 only gives A with real distinct eigenvalues in [-1,1]. Distinct real eigenvalues imply diagonalizability over the reals, not normality, so the lemma does not apply as stated. It's possible a similar super-polynomial upper bound holds for the non-normal case, but the paper doesn't prove it. This is fixable: the authors should either prove the needed version under Assumption 1 or restate the claim.

Who this is for: people working on finite-sample guarantees for subspace identification and output-feedback LTI systems. I would send it to peer review, not desk reject, because the core matrix analysis is competent and the structural questions are worth hashing out. But it needs major revision before acceptance: fix the upper-bound/lower-bound conflation and close the Lemma 6 gap.

Referee Report

3 major / 4 minor

Summary. The paper studies finite-sample error bounds for subspace identification of discrete-time LTI stochastic systems with no external inputs. It derives high-probability upper bounds on the identification errors of A, C, and the Kalman gain K (Theorems 1 and 2), an upper bound on the decay of the smallest singular value of the block Hankel matrix (Lemma 2), and a bound on the Hausdorff distance between the true and estimated system poles (Theorem 3). The abstract and Section 6 further claim a super-polynomial sample size requirement in n/m for constant-error identification, expressed in eqs. (32) and (33).

Significance. The paper contains a genuinely useful concentration analysis for the block Hankel estimate and a clean robustness lemma for subspace identification; those parts are presented with explicit constants and detailed proofs. If the advertised sample-complexity and pole-rate results were valid, they would be a meaningful contribution to the finite-sample system identification literature. However, the central 'super-polynomial sample size is required' claim is not actually proved, and the singular-value lemma that drives it is applied outside its stated assumptions. The strength of the paper therefore rests on the Hankel-error and perturbation bounds, which may be salvageable, but the headline contributions as stated are not established.

major comments (3)

[Section 3, eqs. (32)-(33); also Abstract and Section 6] The derivation of eqs. (32) and (33) is logically invalid. These equations are obtained by inverting the upper bounds of Theorem 2 and Lemma 2 and then concluding that N 'must satisfy' a lower bound of the displayed order. An upper bound on the estimation error of the form error <= f(N) implies a sufficient condition on N for the error to be below a target threshold; it does not imply that N has to be that large. No lower bound on the SIM estimation error, and no minimax or information-theoretic lower bound, is proved anywhere in the manuscript. Therefore the abstract's claim that constant-error identification 'requires' a super-polynomial sample size in n/m, and the analogous wording in Section 6, are unsupported. The authors should reframe these statements as sufficient sample-size conditions, or provide a genuine lower-bound argument.
[Appendix 6.4, Lemma 2 proof; Appendix 6.1, Lemma 6] The proof of Lemma 2 applies Lemma 6 to the extended observability matrix Gamma_T, but Lemma 6 requires the base matrix D to be unitary diagonalizable with real eigenvalues. Assumption 1 only requires A to have real distinct eigenvalues in [-1,1] and does not imply that A, or the matrix D arising in the Krylov structure, is normal or unitary diagonalizable. Consequently the inequality sigma_n(Gamma_T) <= 4 rho^{-...} log(2mT) ||Gamma_T|| is not justified under the assumptions of the paper. Since Lemma 2 is used to obtain sigma_n(H_T) <= ... and hence eqs. (31)-(33), the super-polynomial decay of sigma_n(H_T) and all sample-complexity conclusions depending on it are unproved.
[Section 4 and Theorem 3] The advertised pole-rate O(N^{-1/(2n)}) is a corollary of Theorem 3 and Lemma 2 through the quantity Delta in eq. (38). Because Lemma 2 is not established under Assumption 1 (see the previous comment), the pole-rate claim is not established either. Additionally, even if Lemma 2 were available, the rate statement uses eq. (37) with Delta containing T^5 and sigma_n(H_T); the paper does not verify that the condition of Theorem 3, namely the perturbation bound ||H_T - hat H_T|| <= sigma_n(H_T)/4, is satisfied with high probability by the sample bound of Theorem 1. This is a conditional statement, but the paper presents it as an unconditional rate result.

minor comments (4)

[Appendix 6.1, Lemma 3] The statement of Lemma 3 introduces a matrix M but the inequalities are written for A; please replace M with A for consistency.
[Section 2.1, eq. (19)] The notation hat K_T(:, -m:-1) for the last m columns should be explicitly defined, since negative-index slicing is not standard in all readers' conventions.
[Abstract and Section 6] The phrase 'decreases at a rate of at least O(...)' is misleading: an upper bound of order O(1/sqrt N) means the error decreases no slower than 1/sqrt N, so the standard phrasing is 'at most O(...)' or 'with high probability bounded by O(...)'.
[Section 5, Figure 2] The caption's subfigure indices are confusing: the text refers to subfigures (b) and (e) for pole errors, but the caption lists the pole error panels as (b) and (e) after (a) and (d); please check that the references in the text match the actual arrangement.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; the super-polynomial claim rests on an independent though imported lemma, and the 'requires' wording overstates a sufficient-condition bound.

full rationale

The claimed derivation is not circular. The error bounds in Theorem 1 are proved from Gaussian concentration inequalities (Corollary 1, Lemma 4), norm bounds (Lemma 7), and standard perturbation arguments (Lemma 1, Theorem 2, Theorem 3); no parameter is fitted to data and no target quantity is used as an input to its own derivation. The super-polynomial sample-size conclusion depends on Lemma 2, whose proof imports Lemma 6 from the same first author's prior work [45]. That is a self-citation, and it is load-bearing for the n/m dependence. However, Lemma 6 is a parameter-free general bound on Krylov-matrix singular values with stated assumptions that do not include the present identification error or sample complexity, so under the review rules it counts as independent evidence rather than circularity. The paper does overstate (32)-(33) as necessary sample sizes ('must satisfy', 'requires') when the proof only establishes sufficient conditions obtained by inverting upper bounds; this is a direction-of-implication overclaim, not a circular reduction. Likewise, the mismatch between Lemma 6's unitary-diagonalizable assumption and Assumption 1 (real distinct eigenvalues in [-1,1] do not imply normality) is a correctness and verification gap in the applicability of the imported lemma, not a circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The derivation introduces no new physical entities and fits no constants to data. Its load-bearing assumptions are the usual minimality and steady-state conditions plus an imported singular-value lemma whose stated hypothesis is stronger than Assumption 1.

free parameters (1)

Trajectory half-length T = T ~ Theta(log N)
Algorithmic parameter chosen by hand to balance the cross-term error and the Kalman filter truncation bias. The claimed rates ignore log factors under this scaling, and the validity of the perturbation condition depends on it.

assumptions (5)

domain assumption Assumption 1: (A,C) observable, (A,Q^{1/2}) controllable, and A stable or marginally stable with real distinct eigenvalues in [-1,1].
Defines the system class and is needed for Kalman filter existence and for the real-eigenvalue condition used in Lemma 6. It does not imply the normality needed by Lemma 6.
domain assumption Assumption 2: The innovation form has converged to steady state, with xhat_0 ~ N(0,P).
Needed for the i.i.d. Gaussian past outputs and covariance computations in Lemma 8.
domain assumption Assumption 3: The order n of the innovation form is known.
The SIM rank-n truncation requires the state dimension n as an input.
ad hoc to paper Lemma 6: A smallest singular value bound for a Krylov matrix whose base matrix D is unitary diagonalizable with real eigenvalues.
Stated without proof and imported from the authors' prior work arXiv:2310.11790. It is used in Lemma 2 to conclude super-polynomial decay of sigma_n(H_T), but its hypothesis is not matched by Assumption 1.
standard math Gaussian concentration lemmas (Lemma 3, Corollary 1, Lemma 4).
Used for the invertibility of Y_p Y_p^T and for the cross-term and truncation-bias bounds; these are standard results.

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Cite this review

Pith. "Pith review of Finite Sample Analysis of Subspace Identification for Stochastic Systems." pith.science (2026). https://pith.science/paper/HVNMKBOO

@misc{pith2026250118853,
  author       = {Pith},
  title        = {Pith review of: Finite Sample Analysis of Subspace Identification for Stochastic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HVNMKBOO}},
  note         = {Machine review of arXiv:2501.18853}
}

abstract

The subspace identification method (SIM) has become a widely adopted approach for the identification of discrete-time linear time-invariant (LTI) systems. In this paper, we derive finite sample high-probability error bounds for the system matrices $A,C$, the Kalman filter gain $K$ and the estimation of system poles. Specifically, we demonstrate that, ignoring the logarithmic factors, for an $n$-dimensional LTI system with no external inputs, the estimation error of these matrices decreases at a rate of at least $ \mathcal{O}(\sqrt{1/N}) $, while the estimation error of the system poles decays at a rate of at least $ \mathcal{O}(N^{-1/2n}) $, where $ N $ represents the number of sample trajectories. Furthermore, we reveal that achieving a constant estimation error requires a super-polynomial sample size in $n/m $, where $n/m$ denotes the state-to-output dimension ratio. Finally, numerical experiments are conducted to validate the non-asymptotic results.

Figures

Figures reproduced from arXiv: 2501.18853 by the authors.

**Figure 1.** A two-mass spring-damper system. Let q1(t) and q2(t) denote the horizontal displacements of the masses m1 and m2, respectively. These displacements constitute the system outputs. Applying Newton’s second law yields the following equations of motion: m1q¨1 = −k1q1 + k2(q2 − q1) − c1q˙1 + c2(˙q2 − q˙1), m2q¨2 = −k2(q2 − q1) − k3q2 − c2(˙q2 − q˙1). Defining the state vector and output vector as x(t) = h q1(t) ˙q1(t) q… view at source ↗

**Figure 2.** Subfigures (a) and (d): The estimation error of the block Hankel matrix [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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