Weak and Strong Data Informativity with Structural Priors for System Analysis

Keke Huang; Keyou You; Wenxuan Xu; Yuan Zhang; Yun Feng

REVIEW

Weak and Strong Data Informativity with Structural Priors for System Analysis

Yuan Zhang, Wenxuan Xu, Yun Feng, Keke Huang, Keyou You

Systems and Control · Submitted 28 Oct 2026

Pith's verdict: Weak versus strong data informativity with affine structural priors is a genuine extension of van Waarde-style informativity, and the paper's main rank-test characterizations survive scrutiny despite a few proof gaps.

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arXiv ID2610.38459 (version 2)

Pith IDGOZ56IZF, Pith's permanent record of this paper and its review

CitationKeke Huang, Keyou You, Wenxuan Xu, Yuan Zhang, Yun Feng. "Weak and Strong Data Informativity with Structural Priors for System Analysis." arXiv:2610.38459 (2026). https://pith.science/paper/GOZ56IZF

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Editor's note

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I read this one carefully. The core idea is real: in existing informativity theory, a dataset either certifies a property for all consistent systems or is useless, and the paper shows that by asking the existential question (at least one consistent system has the property) you can still certify controllability, stabilizability, stability, or their complements from data that fail the standard rank condition. The weak/strong distinction, the generic dichotomy for controllability, and the affine pseudoinverse parameterization are the actual contributions. Example 4 makes the main practical point honestly: with a single structural prior on the fourth column of A, T=3 samples certify controllability even though T

The paper is also more careful than the average submission in this area. Consistency conditions for data and priors are stated explicitly and used before the parameterizations. Remark 2 and Proposition 6 give useful consistency checks against known strong-informativity results. The authors acknowledge the main limitations themselves: the prior must be valid for the true system, the noisy-data extension only covers noise in a known subspace, general affine priors are computationally hard, and Theorem 4's exact condition quantifies over all lambda and v rather than providing a finite algorithm. Those are scope limits, not hidden flaws.

The soft spots are real but, as far as I can tell, not load-bearing. Corollary 2 is stated without proof and the authors call it obvious; the stabilizability boundary can have subtleties with eigenvalues on the unit circle, so they should either provide the proof or explicitly say it follows line by line from Theorem 4. Corollary 4's sufficiency step invokes a rectangular descriptor PBH test from [29,30]; the rank reduction in the proof looks plausible, but the hypotheses of that PBH variant need to be checked or the argument expanded. Finally, the NP-hardness claims for stability and controllability under general affine priors rely on two arXiv preprints [24,25] rather than peer-reviewed sources. That is a minor issue for a theory paper, but it should be cleaned up before publication.

I found no internal contradiction in the main results and no sign of curve-fitting. The free parameters are genuinely free, and the conclusions follow from the algebra. The paper deserves a serious referee. I would send it to review with a request to fill the Corollary 2 gap and verify the descriptor PBH invocation; both appear fixable. It belongs in a systems/control journal at the level of Automatica or IEEE TAC, and it will likely be cited by anyone working on data informativity with structural knowledge.

Referee report

Significance

The framing is sensible and the paper is unusually honest about its own limits: the NP-hardness results (Propositions 3 and 5) usefully delimit the tractable class instead of overclaiming, the negative noise results in Corollaries 5–6 are reported rather than hidden, and §5 states explicitly that the exact condition (25) "does not itself provide an efficient finite verification procedure." The single-affine parameterization (21)–(24) is clean and reduces to a rank-one-parameter structure to which the Corfmat–Morse criterion applies, which is the right technical move. The appendix contains complete proofs for Theorems 1, 4, 5, Propositions 1, 4, 6 and Corollaries 1, 4. I checked the numerical data in Examples 2, 3 and 4 against the stated A_u, B_u and they are exactly consistent, which is a good sign for reproducibility. If the characterizations are correct as stated, the framework gives a systematic way to extract conclusions from datasets that fail the van Waarde condition, and the prior-relaxation claim would be of real interest.

Major objections (3)

[§5 (text following Theorem 4) and Example 4] The reduction claimed for S_l = I_n and the verification offered in Example 4 appear to contradict each other. The paper states that when S_l = I_n, Condition 2 of Theorem 4 reduces to rank[λX_p − X_f, λE_nS_r − Q] = n for all λ ∈ C. In Example 4, S_r = e_4 and Q = [0,0,0.1,1]^T, so E_4S_r = e_4 and the reduced test becomes rank[λX_p − X_f, λe_4 − Q] = 4 for every λ ∈ C, i.e. the 4×4 matrix N(λ) = [λx(0) − x(1), λx(1) − x(2), λx(2) − x(3), λe_4 − Q] must be nonsingular for every λ. Expanding in the four λ-affine columns with the reported data (x(0) = [0,−1,0,0.5]^T, x(1) = [−0.1,−1.2,0.05,0.6]^T, x(2) = [−0.22,−1.4352,0.11,0.6196]^T) gives, to the accuracy of hand computation, det N(λ) ≈ −9.8×10⁻⁴ λ² − 4×10⁻⁵: the λ⁴ and λ³ coefficients cancel exactly (det[x(0),x(1),x(2),e_4] = 0 and −det[x(0),x(1),x(3),e_4] = det[x(0),x(1),x(2),Q] = −9.8×10⁻⁴), while the λ² coefficient det[x(0),x(1),x(3),Q] − det[x(0),x(2),x(3),Q] ≈ −9.75×10⁻⁴ is far from zero. If this is right, det N(λ) has complex roots near λ ≈ ±0.2i, and the reduced condition fails for Example 4's data. Either the reduction stated after Theorem 4 is not valid as written, or Example 4 does not satisfy Condition 2; both cannot stand. Please resolve this and, because Condition 2 must hold for all λ ∈ C, report the exact determinant polynomial of the minor actually used (for a 4×4 minor of a λ-affine matrix this polynomial has degree at most 4 in λ, so "nonzero for all λ ∈ C" is a claim that it is a nonzero constant). This matters beyond the example, since the statement that strong informativity "can be satisfied from fewer data, possibly T < n" rests entirely on this example.
[§5, Theorem 4 and the paragraph following it] Theorem 4 states an exact characterization (25) that quantifies over all λ ∈ C and all v ∈ K_λ \ {0}, and the paper itself notes that this "does not itself provide an efficient finite verification procedure"; the two rank conditions offered as sufficient become necessary and sufficient only when S_l = I_n, and even then the reader is not told how the condition "for all λ ∈ C" is checked finitely. Combined with the preceding comment, a reader cannot identify which displayed condition is the actual algorithm and what it costs. Please (i) give a decision procedure or complexity statement for (25) — for instance by eliminating λ from the real linear feasibility problem (41), or by reducing to a finite generalized-eigenvalue computation — and (ii) supply the derivation of the S_l = I_n reduction from Condition 2 to rank[λX_p − X_f, λE_nS_r − Q] = n, which is currently asserted without proof. It would also help to say once, for Theorems 1 and 4 and Corollaries 1–3, how a rank condition over all λ ∈ C is verified in finite time (Kronecker form or generalized eigenvalues), so that the abstract's claim that verification "reduces to rank tests of matrix pencils" has concrete computational content. The authors acknowledge this: “Although each fixed (λ, v) requires only a real linear consistency test, the condition quantifies over all complex eigenvaluesλand candidate left vectorsvinK λ and does not itself provide an efficient finite verification procedure.”
[Appendix, proof of Corollary 1, Case 1] The sufficiency argument in Case 1 (H ≢ 0) moves from "for each fixed λ there is some Y" to "a generic choice of Y works simultaneously for all λ with |λ| ≥ 1" via the sentence "From [19], the set of parameters Y that result in uncontrollability at any non-fixed mode constitutes a proper algebraic variety." As written this set is a union over uncountably many λ, and such a union of proper subvarieties need not be proper, so the step needs justification: either an argument using the fact that the number of uncontrollable modes of (A_Y, B_Y) is bounded by n (so the bad set is the projection of a semialgebraic set of bounded complexity, and can be shown to be proper by exhibiting one Y and invoking openness of the full-rank condition on a compact set), or a precise pointer to the statement in [19] being used and its hypotheses. Corollary 1 is one of the paper's three weak-informativity characterizations, and Case 1 covers the non-degenerate situation, so this gap is load-bearing for the claim that weak stabilizability is decidable by (13).

Minor objections (8)

[Notations (§2)] The paragraph defines M^⊥ = I − MM^† and also M^⊥ = I − M^†M with the same symbol; the two objects are used crucially later (Z_p1^⊥ versus Z_p^⊥, S_l^⊥, Σ_1^⊥, Σ_2^⊥, E^⊥), and the reader cannot tell from the text which is meant at each occurrence. Please introduce distinct symbols for the left and right complementary projectors and use them consistently.
[Example 4] The sentence "since rank([X_p^T U_p^T]^T) = 3 and rank([Z_p S_r]) = 4 < 5" cannot be right as written, because Z_p ∈ R^{5×3} and S_r ∈ R^{5×1}, so Z_p S_r is not conformable; presumably a stacked matrix such as [Z_p^T, S_r^T]^T was intended. Please state the matrix whose rank is 4 and explain why it shows that [A B] is not uniquely determined.
[§3.3, Fact and Remark 1; Introduction, contribution (1)] The generic dichotomy is listed as the first contribution, but its proof is omitted as standard ("see, e.g., [21,19,22]"). This is acceptable as a citation, but the introduction and abstract should then describe the dichotomy as following from the structural-controllability theory rather than as established here, or a three-line proof via the Zariski openness of controllability should be included, together with a note that "almost all" is with respect to the measure on the affine image of Y (the parameterization is not injective when Z_p^⊥ has a kernel).
[§4.2, closing paragraph] The claim that verifying weak informativity for stability under single-affine priors is "at least as hard as the SOF problem" is supported only by the informal construction "one can suitably choose U, X such that Σ_2^⊥Z_p^⊥ = 0 while Σ_1^⊥ and Z_p^⊥ are general enough." Either formalize the reduction (specifying the choice of Z_p and of the prior that encodes A_0 + B_0KC_0) or label the statement as an observation or conjecture.
[Appendix, proof of Theorem 5] The proof passes from "H(z) must be identically zero" to "CA_0^kS_l^⊥ = 0 for all k ≥ 0" and then identifies this with Condition (3), which is imposed only for k = 0,...,n−1. The reduction should be stated explicitly via the Cayley–Hamilton theorem, and the sentence "for any z_0 > 1 with H(z_0) ≠ 0, there exists a real U" deserves the one-line justification that a rank-one U = v w^T with w^T H(z_0) v = 1 exists whenever H(z_0) ≠ 0.
[§6, Corollaries 5–6] The noise model is explicitly taken from [28] (Remark 5). Please state explicitly whether the conclusion that strong informativity is possible only when D = {0} is new relative to [28], or is a restatement of a result there; if the latter, the contribution should be described accordingly. Also, the abstract's "extended to a class of noisy data" would read more accurately if it said that, for subspace noise, only weak informativity survives and strong informativity collapses to the noiseless case.
[References [24], [25]] The NP-hardness of static output feedback stabilization, used in the proof of Proposition 3, is attributed only to two 2026 arXiv preprints. The classical complexity literature on this and related problems (for instance the results summarized in [26] and the well-known NP-hardness papers on simultaneous stabilization and static output feedback) should also be cited so the reader can locate a peer-reviewed source.
[§2.2 (validity of the prior) and §1 (interpretation of weak informativity)] The assumption that the prior is exactly valid is stated clearly, but since the paper's selling point is certification, a short remark on the consequences of a misspecified prior (the consistent set may exclude the true system, so neither weak nor strong informativity transfers) would help the reader judge the scope. Relatedly, for controllability the dichotomy implies that weak informativity amounts to "not all consistent systems are uncontrollable"; stating this plainly in §3.3 would prevent readers from over-reading weak informativity as a positive controllability certificate.

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The paper

The paper claims

The paper argues that data too short for the standard informativity test can still certify controllability or stability once affine structural priors are available, via rank tests on matrix pencils.

The reading

Standard data informativity asks whether every system consistent with the measured data has a target property, and its rank test $\mathrm{rank}(\lambda X_p - X_f) = n$ frequently fails when experiments are short. This paper argues that such a failure is ambiguous: it does not distinguish “every consistent system violates the property” from “at least one consistent system possesses it.” The paper therefore splits the notion into weak informativity (at least one consistent system has the property) and strong informativity (every one does), admits prior structural knowledge as affine constraints on the system matrices, and shows that in the tractable cases, notably single-affine priors such as fixing a known block of $[A, B]$, both notions reduce to rank tests on matrix pencils built from the data and the prior. The payoff claimed is that datasets conventionally discarded as non-informative can still certify controllability, stabilizability, or stability, and that a structural prior can lower the excitation requirement to fewer samples than the state dimension, as in Example 4 with $T=3 < n=4$.

What carries the argument

The carrying machinery is the Moore–Penrose parameterization of the data-consistent set, $[A, B] = X_f Z_p^\dagger + Y(I - Z_p Z_p^\dagger)$, combined with the structural-controllability criterion stated as Lemma 1, which tests an affinely parameterized family for the existence of a controllable realization through a graph condition and rank conditions at eigenvalues. For single-affine priors $S_\ell [A, B] S_r = Q$, free parameters enter through rank-one coefficients, which keeps the tests polynomial-time; general affine priors produce matrix nets whose structural-controllability checking is exponential, and weak informativity for stability under them is NP-hard. The strong-informativity rank tests assemble pencils such as $\begin{bmatrix} \lambda X_p - X_f & S_\ell^\perp \end{bmatrix}$ and $\begin{bmatrix} \lambda X_p - X_f & (\lambda E_n - M_0) Z_p^\perp W \end{bmatrix}$, where the block contributed by the prior supplies rank that short data alone cannot.

Extended reading notes

Core claim

By parameterizing every system consistent with the data as $[A_Y, B_Y] = X_f Z_p^\dagger + Y(I - Z_p Z_p^\dagger)$ with a free matrix $Y$, the paper converts the data-consistent set into an affine family and then into a linearly parameterized system to which structural-controllability tests apply. This yields necessary and sufficient conditions for weak informativity for controllability, stabilizability, and stability without priors; a dichotomy for controllability in which either no consistent system is controllable or almost all are; explicit parameterizations of the jointly consistent sets under general affine priors; and rank conditions for strong informativity for controllability and stability under single-affine priors. The paper further shows that in the noisy setting strong informativity for controllability and stability is attainable only when the data are noiseless, while weak informativity remains serviceable when the noise lies in a known subspace.

Load-bearing premise

Every certificate rests on the assumptions, stated in Sections 2.1 and 2.2, that the data obey the noiseless equation $X_f = [A, B] Z_p$ exactly (or that noise is confined to a known subspace) and that the affine prior is exactly valid for the true system, so a misspecified prior or any noise leaving the subspace can place the true system outside the set being certified.

Editorial extensions

If this is right

Datasets that fail the standard test $\mathrm{rank}(\lambda X_p - X_f) = n$ keep their value: weak informativity says whether the property is compatible with the data, and its failure certifies the complementary property.
Single-affine priors lower excitation requirements: Example 4 certifies controllability of a four-state inverted-pendulum model from $T=3$ samples, fewer than the state dimension, using prior knowledge of one column of $A$.
Because controllability is generic in the consistent set whenever it is possible, a weak certificate is not a fragile existence claim: almost all systems consistent with the data will be controllable.
With noise restricted to a known subspace, weak informativity still yields conclusions about controllability and stability, but strong informativity collapses to the noiseless case, so certified analysis of noisy data must use weak informativity or richer noise models.

Reading between the lines

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

Direct numerical checks of the paper's central selling point are easy to build: generate random datasets with $T < n$, add random single-affine priors, count how often the new rank tests certify controllability when $\mathrm{rank}(\lambda X_p - X_f) = n$ fails, and compare the required sample count with the number of prior-fixed entries.
The weak/strong split is a general template: any property with a PBH-style or rank characterization, such as observability of the dual pair or output controllability, could be certified from short data by the same parameterization whenever the prior keeps the consistent set affine.
The noiseless collapse of strong informativity suggests that for noisy certification the safer design is to enlarge the prior set rather than the noise set, for instance with set-membership bounds on the disturbance as the authors flag for future work, so the certified set provably contains the true system.
Because failure of weak informativity certifies the complement, short experiments could serve as cheap screening tests: a dataset that is not weakly informative for controllability saves the cost of longer experiments aimed at verifying it.

Original abstract

When measurement data are insufficient for unique system identification, data informativity evaluates whether a dataset determines a property across all systems consistent with the data. This paper introduces a unified framework distinguishing between weak and strong data informativity under prior structural knowledge described by affine constraints: data are weakly informative if at least one system consistent with the data and prior knowledge possesses the target property, and strongly informative if every consistent system satisfies it. The latter is consistent with the notion of data informativity in the existing literature. By explicitly parameterizing all consistent systems as affine functions of free parameters via the Moore-Penrose pseudoinverse, we characterize these notions for fundamental system properties-including controllability, stabilizability, and stability. This formulation establishes that verifying weak and strong informativity, when tractable, reduces to rank tests of matrix pencils constructed from the data and prior knowledge. Our findings show that structural priors can lower data-excitation requirements and enable reliable analysis even when data fail to be strongly informative. Finally, the framework is extended to a class of noisy data.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 5,621 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.61 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.43 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

When measurement data are insufficient for unique system identification, data informativity evaluates whether a dataset determines a property across all systems consistent with the data. This paper introduces a unified framework distinguishing between weak and strong data informativity under prior structural knowledge described by affine constraints: data are weakly informative if at least one system consistent with the data and prior knowledge possesses the target property, and strongly informative if every consistent system satisfies it. The latter is consistent with the notion of data informativity in the existing literature. By explicitly parameterizing all consistent systems as affine functions of free parameters via the Moore-Penrose pseudoinverse, we characterize these notions for fundamental system properties-including controllability, stabilizability, and stability. This formulation establishes that verifying weak and strong informativity, when tractable, reduces to rank tests of matrix pencils constructed from the data and prior knowledge. Our findings show that structural priors can lower data-excitation requirements and enable reliable analysis even when data fail to be strongly informative. Finally, the framework is extended to a class of noisy data.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	When measurement data are insufficient for unique system identification, data informativity evaluates whether a dataset determines a property across all systems consistent with the data.	background 1.00	0.06	0.04	asserted 1.00
s1	This paper introduces a unified framework distinguishing between weak and strong data informativity under prior structural knowledge described by affine constraints: data are weakly informative if at least one system consistent with the data and prior knowledge possesses the target property, and strongly informative if every consistent system satisfies it.	contribution 1.00	0.93	0.22	asserted 1.00
s2	The latter is consistent with the notion of data informativity in the existing literature.	interpretation 0.59	0.06	0.03	hedged possibility 0.82
s3	By explicitly parameterizing all consistent systems as affine functions of free parameters via the Moore-Penrose pseudoinverse, we characterize these notions for fundamental system properties-including controllability, stabilizability, and stability.	method 0.93	0.95	0.08	asserted 1.00
s4	This formulation establishes that verifying weak and strong informativity, when tractable, reduces to rank tests of matrix pencils constructed from the data and prior knowledge.	result 0.71	0.91	0.18	asserted 0.97
s5	Our findings show that structural priors can lower data-excitation requirements and enable reliable analysis even when data fail to be strongly informative.	result 0.61	0.94	0.67	asserted 0.98
s6	Finally, the framework is extended to a class of noisy data.	contribution 0.96	0.88	0.04	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.77 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Strong and weak data informativity without priors.

Figures reproduced from arXiv:2610.38459 by the authors.

Discussion

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