{"id":"0f197b80-3f51-4e0d-9f9a-c860a531512c","arxiv_id":"1908.01039","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The eigenvalues of an unknown linear dynamical system's state-transition matrix can be consistently estimated from output time series by fitting the autoregressive parameters of an ARMA model, at a root-T convergence rate.","lead":"This paper shows that the eigenvalues of the transition matrix in a linear dynamical system can be estimated from time series output without estimating all system parameters, using the connection between linear systems and ARMA models. The result enables fast clustering of time series by their underlying dynamics, with tests on synthetic data and ECG signals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The advertised O(T^{-1/2}) eigenvalue convergence rate is only valid for simple spectra; the paper's own Theorem 4.2 gives O(epsilon^{1/n}) for repeated eigenvalues, and Theorem 4.2's displayed polynomial is the reciprocal AR polynomial, whose roots are not the eigenvalues.","rationale":"I read the paper as establishing a valuable and mostly correct connection between LDS eigenspectra and AR parameters of ARMA representations. Theorem 4.1's identification claim is sound under the stated observability and no-common-factor assumptions, and the iterated regression approach is a sensible, fast estimator. The reader's conditional verdict is appropriate. However, I do not think the strongest load-bearing weakness is the unproved regularization consistency: a fixed L2 penalty on the nuisance MA coefficients is asymptotically negligible relative to the O(T) sum of squares, so the claim that regularization does not change the asymptotic distribution is plausible and likely correct. The more serious issue is that the paper advertises a T^{-1/2} eigenvalue convergence rate in the Introduction and Section 6.3, while Theorem 4.2's own bound degrades to O(epsilon^{1/n}) for repeated eigenvalues; the supported general rate is O_p(T^{-1/(2n)}). This is an internal inconsistency, not a disagreement with external consensus, and it affects the headline guarantee. The related error in Theorem 4.2's polynomial—roots of the reciprocal AR polynomial are reciprocals of eigenvalues—compounds the issue, since a literal reading of the theorem gives the wrong convergence target. Both issues are fixable without changing the main conceptual contribution, which is why I would keep the verdict CONDITIONAL rather than REJECT. The concrete simulation with a Jordan block would settle whether the slower rate actually occurs and would also expose the reciprocal-root misstatement.","tokens_in":22178,"tokens_out":19822,"duration_ms":196757,"concrete_test":"Simulate a two-dimensional LDS with A = [[0.5, 1], [0, 0.5]] (double eigenvalue 0.5), C = I, B = I, output noise xi ~ N(0, 0.01), for T = 10^3, 10^4, 10^5. Run Algorithm 1 with n = 2 and compute the maximum eigenvalue error from the roots of z^2 - phi_hat_1 z - phi_hat_2. If the log-log slope is approximately -1/4 rather than -1/2, the unqualified O(T^{-1/2}) rate claim fails for repeated spectra. Also record roots of 1 - phi_hat_1 z - phi_hat_2 z^2: they should approach 2 rather than 0.5, confirming that Theorem 4.2's displayed polynomial is the reciprocal polynomial.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central quantitative claim—that eigenvalues are estimated at rate O_p(T^{-1/2})—is not supported by the paper's own theorems. Theorem 5.1 gives AR parameter error epsilon = O_p(T^{-1/2}). Theorem 4.2 then says the eigenvalue error is O(epsilon^{1/n}) in general and O(epsilon) only when all eigenvalues are simple. For a repeated eigenvalue of multiplicity m, root splitting gives O(epsilon^{1/m}), so with m > 1 the eigenvalue rate is strictly worse than T^{-1/2}. The Introduction and Section 6.3 state the O_p(T^{-1/2}) eigenvalue rate without any simplicity assumption, and the experiments use random matrices, which have simple eigenvalues almost surely, so the advertised guarantee is untested for the non-generic but permitted repeated-spectrum case. Separately, Theorem 4.2 defines r_i as roots of the lag polynomial 1 - phi_1 z - ... - phi_n z^n, which is L^n chi_A(L^{-1}); its roots are reciprocals of the eigenvalues of A, not the eigenvalues. As written, the theorem's statement that these roots converge to the true eigenvalues is literally false. The fixes are local—state the simplicity assumption and use the monic polynomial z^n - phi_1 z^{n-1} - ... - phi_n—but the correctness of the headline claim depends on them.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a method to estimate the eigenvalues of the state-transition matrix A of a linear dynamical system (LDS) from output time series alone, without full system identification. The theoretical core is a correspondence between LDSs and ARMA/ARMAX models: the autoregressive parameters of the output series recover the characteristic polynomial of A (Theorem 4.1), and small AR-parameter error implies small eigenvalue error, with a general O(epsilon^{1/n}) bound and an O(epsilon) bound under simplicity (Theorems 4.2 and 4.3). The authors combine this with the consistent iterated-regression estimator of Tsay and Tiao (1984), adding L2 regularization on the MA coefficients, to obtain a practical algorithm with an advertised O_p(T^{-1/2}) eigenvalue convergence rate. The method is applied to time series clustering, with experiments on synthetic LDS data and real ECG signals showing competitive clustering quality and much lower runtime than EM-based LDS baselines.","tokens_in":22433,"tokens_out":4960,"duration_ms":48799,"significance":"If the main claims hold, the paper makes a useful conceptual point: meaningful information about linear dynamics (the eigenspectrum) can be learned from outputs without solving the hard full-identification problem. The model equivalence is standard but the paper gives explicit perturbation bounds and a computationally simple algorithm with code. The clustering application is natural and the experiments are reasonable. However, the advertised convergence rate is only proven under a simplicity assumption, and one theorem statement contains a reciprocal-polynomial error; these are local but load-bearing issues that the authors should fix.","major_comments":[{"comment":"The statement defines r_i as the roots of the polynomial 1 - phi_hat_1 z - ... - phi_hat_n z^n. This is the reciprocal lag polynomial, whose roots are the reciprocals of the eigenvalues (since the characteristic polynomial is lambda^n - phi_1 lambda^{n-1} - ... - phi_n). As written, the claim that these roots converge to the true eigenvalues is literally false. The monic polynomial z^n - phi_hat_1 z^{n-1} - ... - phi_hat_n should be used instead.","section":"Theorem 4.2"},{"comment":"The paper repeatedly states that eigenvalues are estimated at rate O_p(T^{-1/2}) without qualification. Theorem 4.2 gives O(epsilon) only when all eigenvalues are simple; in the general case the rate is O(epsilon^{1/n}), which combined with Theorem 5.1 gives only O_p(T^{-1/(2n)}). The unqualified statements in the abstract, introduction, and Section 6.3 are therefore not supported by the theorems. The simplicity assumption should be stated wherever the O_p(T^{-1/2}) eigenvalue rate is claimed.","section":"Abstract, Section 1, Section 6.3"},{"comment":"The claim that adding L2 regularization to the MA coefficients 'does not change the asymptotic property of the estimator' is asserted without proof or citation. Theorem 5.1 covers the unregularized iterated regression of Tsay and Tiao (1984), not Algorithm 1 as proposed. Since the paper's consistency and rate claims apply to the regularized algorithm, this gap needs to be filled (or the claim qualified).","section":"Section 5, 'Convergence rate'"},{"comment":"The theorem as stated claims that every output dimension of any LDS is ARMAX(n,n,n-1) with AR parameters recovering the characteristic polynomial. The proof in Appendix A.2 requires the additional assumption that there are no common factors in chi^dagger_A and Gamma (no pole-zero cancellation) to conclude that chi^dagger_A is the AR polynomial. Without this assumption the output series can be generated by a lower-order ARMA model and the full eigenspectrum of A is not recoverable from the AR parameters. This condition should be part of the theorem statement (and of Corollary 4.1), not only of the proof.","section":"Theorem 4.1 and Appendix A.2"}],"minor_comments":[{"comment":"The companion matrix is defined with one orientation in Definition B.1 and used with a different sign/orientation in the state-space construction in Equation (3) (C(-Phi)^T). This makes the simple-eigenvalue proof harder to follow; the conventions should be reconciled.","section":"Equation (3) and Definition B.1"},{"comment":"The 'if and only if' claim for LDSs of different dimensions relies on the no-cancellation premise; the paper should explicitly connect this to the assumption of Theorem 4.1.","section":"Corollary 4.1"},{"comment":"The claim that multidimensional outputs share the same autoregressive parameters and that averaging per-channel estimates preserves the statistical rate is not justified; a sentence of explanation or a citation would be helpful.","section":"Section 5.1"},{"comment":"Figure 1 is referenced as showing eigenvalue estimation error, but the captioned figure content is not reproduced in the text I reviewed; the authors should ensure the figure is present and readable in the final version.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The core idea is sound and the fixes are local: correct the reciprocal-polynomial statement in Theorem 4.2, qualify the O_p(T^{-1/2}) claims with the simplicity assumption, and either prove or clearly qualify the regularization-preserves-consistency claim. The repeated-spectrum case is not tested in simulations, which use random matrices; if the authors can add a small repeated-eigenvalue experiment it would strengthen the paper. Overall I view the manuscript as close to acceptance after these revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Katie,\n\nQuick take: this paper deserves review. The core idea—estimate the eigenspectrum of an LDS from output alone via the AR parameters of its equivalent ARMA model, skipping full system identification—is sound and practical. The authors credit the classical LDS-ARMA equivalence (Åström–Wittenmark, Kailath) and build on it with something new: an explicit condition-number bound for simple eigenvalues via Vandermonde inversion, and a regularized iterated-regression estimator that is fast and works well. Clustering experiments on synthetic and ECG data are convincing, and the basis-invariance is a real plus when measurement procedures differ across sources.\n\nThe soft spots are real but local. The advertised O_p(T^{-1/2}) eigenvalue rate is only justified for simple spectra. Theorem 4.2 itself gives O(ε^{1/n}) in general and O(ε) for simple eigenvalues; with repeated eigenvalues the rate is strictly worse. The Introduction and Section 6.3 state the fast rate without qualification, and the experiments use random matrices, which have simple eigenvalues almost surely, so the faster rate is untested for repeated spectra. Second, Theorem 4.2 defines r_i as roots of 1 − φ̂_1 z − ... − φ̂_n z^n, which is the reciprocal lag polynomial; its roots are reciprocals of the eigenvalues, not the eigenvalues. As written, that theorem is literally false. The fix is easy—use the monic polynomial z^n − φ̂_1 z^{n−1} − ... − φ̂_n—and Appendix B already works with the monic polynomial, so this is a statement bug, not a proof bug. Still, it should be corrected. Third, the claim that adding regularization preserves O(T^{-1/2}) consistency is asserted without proof. Plausible, but it needs a citation or a proof. Finally, the regularization coefficient for synthetic data is unreported, and the ECG value (0.01) was chosen based on simulation. Disclose it and show sensitivity.\n\nNone of this sinks the central contribution. The method works, the theory is mostly right, and the writing is clear. I'd send this to a serious referee with a request to fix the reciprocal issue and qualify the convergence claims. For the time series/ML crowd, this is worth knowing.\n\nBest","headline":"Useful practical method for LDS eigenvalue estimation and clustering; the headline convergence rate is over-stated, and Theorem 4.2 has a reciprocal-polynomial bug that is easy to fix but should not be left as is.","tokens_in":22982,"tokens_out":5185,"would_cite":true,"duration_ms":44150,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B30","62M10","15A18"],"pacs":[],"model":"deepseek-v4-flash","headline":"Outputs alone reveal eigenvalues of hidden linear dynamics","keywords":["linear dynamical systems","eigenspectrum identification","ARMA models","autoregressive parameters","iterated regression","time series clustering","observability","system identification"],"falsifier":"Run the estimator on output from a deliberately unobservable system, such as one where a tightly damped eigenmode is orthogonal to the measurement matrix C; the recovered spectrum should miss that mode, matching the paper's own observability caveat. Alternatively, generate a system with a double eigenvalue and perturb the fitted AR coefficients by ε; the two estimated roots should separate at rate √ε rather than ε, as the general O($ε^{{1/2}}$) bound predicts.","tokens_in":21962,"feed_emoji":"📈","tokens_out":9295,"duration_ms":83693,"temperature":0.7,"pith_summary":"This paper establishes that the eigenvalues of the state-transition matrix in a linear dynamical system can be identified from the observed output series alone, even when the full system parameters cannot be recovered. The mechanism is an exact correspondence: each dimension of the output obeys an ARMAX model whose autoregressive coefficients are, up to sign, the coefficients of the characteristic polynomial of the transition matrix. A regularized iterated least-squares regression estimates those coefficients consistently, converging at rate O($T^{{-1/2}}$) in the sequence length. Because eigenvalues characterize the dynamics up to change of basis, the fitted autoregressive parameters become a natural, computationally cheap similarity measure for clustering time series, including series of different lengths, with temporal offsets, or recorded by different measurement procedures. The paper demonstrates the approach on simulated linear dynamical system data and on electrocardiogram recordings, where it matches or improves clustering quality relative to existing baselines.","feed_headline":"Outputs alone reveal eigenvalues of hidden linear dynamics","feed_subtitle":"No full system identification needed: eigenvalues are hidden in the output's autoregressive coefficients.","key_machinery":"The reciprocal characteristic polynomial χ†_A(L) = L^n χ_A($L^{{-1}}$), applied as a lag-operator polynomial, is the load-bearing identity. The proof expands the hidden state in a Jordan canonical basis of the adjoint A^*, showing that applying (1−λL)^μ to a projected state leaves only a linear combination of past inputs; multiplying over all eigenvalues gives χ†_A(L) y_t as a linear combination of past inputs plus noise. This is what makes the characteristic polynomial of A literally equal to the autoregressive polynomial of the output series. On the estimation side, the companion matrix of the estimated AR polynomial carries the argument: parameter error becomes matrix perturbation, and eigenvalue perturbation bounds (regular splitting for repeated roots, Vandermonde-based explicit bounds for simple roots) convert AR estimation error into eigenvalue error.","core_discovery":"The central discovery is that the eigenspectrum of the unknown transition matrix A is readable from the output series without identifying B, C, or D. Theorem 4.1 shows that, for an n-dimensional linear dynamical system, every output dimension is generated by an ARMAX(n, n, n−1) model whose autoregressive parameters φ1, ..., φn satisfy χ_A(λ) = λ^n − φ1 $λ^{{n−1}}$ − ⋯ − φn; in the no-input case the model reduces to an ARMA(n, n) process. Consequently, two output series share the same autoregressive parameters if and only if the underlying systems have the same nonzero eigenvalues with the same multiplicities (Corollary 4.1). The paper pairs this with perturbation results: an ε error in the autoregressive parameters leads to O($ε^{{1/n}}$) eigenvalue error in general, and O(ε) when all eigenvalues are simple, with an explicit condition-number bound for the latter case. Joined with a consistent iterated-regression estimator of ARMA autoregressive parameters, this yields an O($T^{{−1/2}}$) estimator of the eigenvalues of A from outputs alone.","pith_inferences":["A testable extension is overparameterization: when the assumed hidden dimension is larger than the true order, Theorem 4.1 predicts the extra eigenvalues appear as zeros in the AR polynomial; one could check whether spurious near-zero roots degrade clustering and whether explicitly truncating them helps.","The same AR-coefficient pipeline could serve as a feature map for anomaly detection or change-point analysis in physiological or financial records, where the dynamics shift and the eigenspectrum is the quantity of interest; that use is beyond the paper's clustering experiments.","For multi-site data collection with different device gains, the basis invariance of the spectrum suggests the fitted AR parameters can be pooled directly; a realistic test would be to record the same heart rhythm with two electrode configurations and confirm the estimated spectra agree.","The converse perturbation bound mentioned in the appendix suggests the AR distance can serve as a proxy for eigenvalue distance in general; a sharp, finite-sample version of that converse would turn the clustering algorithm into a distance-based method with known distortion guarantees."],"forward_implications":["Anyone observing a single output channel of a linear system can estimate its stability and oscillation characteristics — for example, distinguishing a stable node from a stable spiral — without knowing the hidden state dimension's parameters.","Time series recorded by different sensors or measurement procedures but governed by the same underlying dynamics cluster together, because the autoregressive parameters are invariant under changes of basis.","The clustering pipeline inherits the estimator's flexibility: it handles series of different lengths, temporal offsets, scale and shift, multidimensional outputs, and rows with missing lagged values, at O(n^3 T + n^4) cost in sequence length T.","The eigenvalue estimates improve at the parametric rate O(T^{-1/2}), so longer records translate into proportionally tighter knowledge about the dynamics, unlike pure AR fitting whose bias does not vanish with T.","Because the method never attempts full system identification, it sidesteps the local-optima and runtime problems of likelihood-based LDS fitting while retaining comparable clustering quality on the reported experiments."],"supporting_citations":[{"why":"Supplies the iterated-regression estimator whose consistency and O(T^{-1/2}) rate the algorithm inherits for AR parameters in ARMA models.","marker":"[Tsay and Tiao, 1984]"},{"why":"Provides the ARMA summation lemma used to conclude that hidden inputs and output noise do not alter the autoregressive polynomial.","marker":"[Granger and Morris, 1976]"},{"why":"Gives the semisimple eigenvalue perturbation expansion underlying the O(ε) bound for simple spectra.","marker":"[Lancaster et al., 2003]"},{"why":"Provides the polynomial root perturbation bound behind the general O(ε^{1/n}) eigenvalue convergence.","marker":"[Beauzamy, 1999]"},{"why":"Supplies the explicit Vandermonde inverse used to bound the eigenvalue condition number in Theorem 4.3.","marker":"[El-Mikkawy, 2003]"},{"why":"Supports the converse perturbation direction — small eigenvalue distance implies small AR-parameter distance — used with simulated data to validate the clustering distance.","marker":"[Ipsen and Rehman, 2008]"}],"fun_headline_variants":["Eigenvalues from outputs: no system ID needed","Hidden eigenvalues revealed by output data alone","Skip system ID: read eigenvalues straight from outputs","Clustering time series via output-derived eigenvalues","Eigenvalues without identification: a linear shortcut"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the system is observable — every hidden mode leaves a trace in the output — and that no pole-zero cancellation, no hidden cancellation between the AR and MA parts, reduces the effective order; if either fails, the output follows a lower-order ARMA model and the full eigenspectrum of A cannot be recovered.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvalues from outputs: no system ID needed","Hidden eigenvalues revealed by output data alone","Skip system ID: read eigenvalues straight from outputs","Clustering time series via output-derived eigenvalues","Eigenvalues without identification: a linear shortcut"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000165,"raw_usage":{"total_tokens":1222,"prompt_tokens":892,"completion_tokens":330,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":261}},"tokens_in":508,"tokens_out":330,"duration_ms":4068,"temperature":1.0,"reasoning_tokens":261,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:24:58.864056+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the estimator on output from a deliberately unobservable system, such as one where a tightly damped eigenmode is orthogonal to the measurement matrix C; the recovered spectrum should miss that mode, matching the paper's own observability caveat. Alternatively, generate a system with a double eigenvalue and perturb the fitted AR coefficients by ε; the two estimated roots should separate at rate √ε rather than ε, as the general O($ε^{{1/2}}$) bound predicts.","supporting_citations":[],"review_version":1}