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REVIEW 3 major objections 5 minor 1 cited by

Secure Data Reconstruction: A Direct Data-Driven Approach

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read By proving that brute-force search and its convex $\ell_1$ relaxation recover the true trajectory under explicit rank and norm conditions on the offline Hankel matrix, this paper makes secure data reconstruction possible for unknown…

desk verdict The brute-force exact-recovery results are solid, but the l1-relaxation guarantees are vacuous unless the Hankel matrix has full column rank—a restriction the paper never states. read the letter →

arxiv 2502.00436 v2 pith:TNCV2L4C submitted 2025-02-01 eess.SY cs.SY

Jiaqi Yan , Ivan Markovsky , John Lygeros This is my paper · ORCID
classification eess.SYcs.SY
keywords securedatareconstructionbehavioralsystemstheorydata-drivencontrolfalseinjectionHankelmatrixl1minimizationgroupLASSOcyber-physicalsystemsecurity
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 whether a trajectory of an unknown linear time-invariant system can be reconstructed exactly after an adversary corrupts up to $k$ entries, or up to $k$ channels, of the received data, using only an attack-free offline trajectory and no state-space model. It answers yes under quantitative richness conditions on the offline data: if the relevant Hankel matrix keeps full rank after deleting any $2k$ rows, or any $2k$ channel blocks, then the brute-force search algorithms recover the true trajectory whenever they terminate. The paper then proves that convex $\ell_1$ and group-LASSO relaxations of these NP-hard problems are also exact under an $\ell_1$-dominance condition or a rank-plus-contraction condition, making the recovery computationally tractable. This matters because data-driven controllers act directly on measured trajectories, so a safeguard that provably returns the true trajectory prevents corrupted data from translating into unsafe control actions.

What carries the argument

The key object is the block-Hankel matrix $H_L(w_d)$ of an offline attack-free trajectory, together with its minimum critical row set $S^*(H_L(w_d))$ or the channel-structured variant. The behavioral lemma says that when $\operatorname{rank} H_L(w_d) = m(B)L + n(B)$, the image of this matrix is exactly the set of length-$L$ system trajectories, so a candidate reconstruction is legal exactly when it lies in that image. The critical-row-set size then measures how much row deletion the matrix tolerates before losing rank, which is precisely the redundancy needed to overwrite $k$ attacked entries or $k$ attacked channels. The $\ell_1$ problem (22) replaces the combinatorial search by minimizing $\|w - H g\|_1$; Theorem 3 certifies exactness through the inequality $\|H_B v\|_1 > \|H_F v\|_1$ for all nonzero $v$, and Theorem 4 converts this into a checkable contraction condition on the matrix $T$ with $H_F = T H_B$.

What would settle it

Fix any offline trajectory satisfying the generalized persistency of excitation condition but with a minimum critical row set of size at most $2k$, construct the entry attack from Proposition 1 by tampering exactly with a critical set, and run Algorithm 1: if it outputs a trajectory different from the true one, the claimed tightness is confirmed and the theorem's condition is shown necessary in that instance. For the $\ell_1$ claim, find any Hankel matrix and attack set with $\operatorname{rank} H_B = \operatorname{rank} H$ and $\|T\|_1 \ge 1$ where the minimizer of (22) differs from the true trajectory; that would refute the sufficiency of Condition 3 alone.

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Extended reading notes

Core claim

The central claim is that secure trajectory reconstruction for an unknown LTI system can be reduced to a rank-persistence property of the Hankel matrix built from offline data. With the generalized persistency of excitation condition, any length-$L$ trajectory is exactly the image of this Hankel matrix; then, if the minimum critical row set of the Hankel matrix has size at least $2k+1$, deleting any $2k$ rows leaves the column space unchanged, so the brute-force algorithm's solution necessarily equals the true trajectory (Theorem 1). The same reasoning with channel-structured critical row sets gives the channel-attack guarantee (Theorem 2). For computation, recovery via (22), an $\ell_1$ minimization over the coefficient vector $g$, is exact provided the benign rows dominate the malicious rows in $\ell_1$ norm for every nonzero direction (Theorem 3), or provided the rank condition $\operatorname{rank} H_B = \operatorname{rank} H$ and the contraction $\|T\|_1 < 1$ hold (Theorem 4).

Load-bearing premise

The load-bearing premise is that the offline Hankel matrix is rich enough that removing any $2k$ rows or channel blocks never reduces its rank, plus, for the $\ell_1$ relaxation, that the attack rows are dominated by healthy rows ($\|T\|_1 < 1$); the paper notes the first condition is computationally difficult to verify and the second depends on the unknown attack set.

Editorial extensions

If this is right

  • If Condition 1 holds, Algorithm 1 never returns a wrong trajectory: any returned candidate equals the true one, so a practitioner can trust a positive reconstruction without knowing the attack set.
  • Together with Lemma 2, the entry-attack guarantee implies the fundamental limit $k \le p(B)/2$; systems with too few outputs cannot be secured against this attacker model regardless of offline data length.
  • The convex problems (22) and (33) inherit exactness in the regimes of Theorems 3 and 4, so secure reconstruction can run in polynomial time instead of enumerating $\binom{qL}{k}$ or $\binom{q}{k}$ subproblems.
  • The noisy-data extension turns the same machinery into a two-stage procedure of convex fit, residual ranking, and least-squares refinement, so the guarantees extend approximately to measurement noise, as demonstrated on mass-spring-damper and quadruple-tank examples.

Reading between the lines

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

  • The paper leaves implicit that Condition 1 is the data-driven counterpart of $2k$-sparse observability in model-based secure state estimation, suggesting that the same redundancy lower bounds transfer to model-free settings.
  • Because the verification in Remark 4 gives a polynomial-time certificate for the $\ell_1$-dominance condition once an attack set is hypothesized, one could turn (22) into a testable detector by sweeping candidate attack sets and keeping those that pass; the paper does not develop this.
  • The analysis assumes the offline data $w_d$ is attack-free and rich enough; a natural extension not treated here is to robustify against a compromised offline set by combining multiple trajectories or applying the critical-row test to an augmented Hankel matrix.
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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

3 major / 5 minor

Summary. The manuscript considers an unknown discrete-time LTI system in the behavioral framework and addresses reconstruction of an L-length trajectory from corrupted measurements, given an attack-free offline trajectory wd satisfying the generalized persistency of excitation condition. Two attack scenarios are studied: up to k arbitrary entries and up to k compromised channels. The authors formulate recovery as combinatorial optimization, propose brute-force algorithms, and prove exact recovery under critical-row-set conditions (Theorems 1 and 2, Propositions 1 and 2). They then replace the cardinality constraints by l1 and group-LASSO relaxations, claiming exact recovery under condition (24) or Condition 3 with ||T||_1 < 1 (Theorems 3 and 4). A heuristic noisy-data extension and numerical examples on mass-spring-damper and quadruple-tank systems are provided.

Significance. The behavioral, model-free formulation is novel and relevant as a safeguard for data-driven control. The paper's main positive features are that the brute-force recovery theorems are mathematically sound and non-circular, relying directly on [17, Corollary 21]; the tightness constructions in Propositions 1 and 2 are meaningful; code is released; and the numerical comparison with a model-based method illustrates the advantage of not needing model knowledge. However, the convex-relaxation guarantees, which are the advertised computationally efficient contribution, have a serious gap in the overdetermined-data regime. The paper needs a substantive revision before the exact-recovery guarantees for (22) can be accepted as stated.

major comments (3)
  1. [Section V-A, Theorem 3, Eq. (24)] Under the standing generalized persistency condition (1), rank H = mL+n. In the common data-rich case T-L+1 > mL+n, H has a nonzero null vector v, and for that v both H_B v and H_F v vanish, so (24) reads 0 > 0 and cannot hold. Thus the hypothesis of Theorem 3 is unsatisfiable exactly in the overdetermined regime where multiple offline trajectories are available, and the proof of (28) gives equality rather than strict inequality for such v. The theorem needs either an explicit full-column-rank assumption, or a modified condition restricted to v not in Null H together with an argument that trajectory recovery still holds despite non-unique g.
  2. [Section V-A, Theorem 4, proof after Eq. (32)] The displayed chain ||H_F v||_1 = ||T H_B v||_1 <= ||T||_1 ||H_B v||_1 < ||H_B v||_1 for v != 0 is invalid when v is in the null space of H, because it would require 0 < 0. The conclusion of Theorem 4 may be salvageable by decomposing v into null and non-null components, but as written the proof does not establish the theorem. Since Theorem 4 is presented as the main tractable verification route, this gap is load-bearing.
  3. [Section V-A, Condition 3; Section VI] Condition 3 is defined through the unknown attack set C via H_B = H|_{[qL]\C}. The statement in Section VI that Condition 3 'can be easily verified offline since H_L(wd) is provided' conflates offline knowledge of wd with offline knowledge of C. In a security setting, C is adversarial and is not available for verification; the paper should either state explicitly that Condition 3 is an unverifiable sufficient condition or provide verifiable proxy conditions. The same limitation applies to condition (24) in Theorem 3, which also depends on C.
minor comments (5)
  1. [Section VI, Fig. 4 caption and text] The phrase 'given in in Fig. 4' contains a duplicated 'in'.
  2. [Section VI-A, comparison paragraph] The citation [Mao2022] does not match any numbered reference in the bibliography; it should be replaced with the corresponding entry (likely [6]) or listed properly.
  3. [Section VI, numerical example parameters] The example uses T=11, L=3, m=1, n=6, so N=9=mL+n; this is precisely the full-column-rank case and therefore avoids the nullspace obstruction discussed above. The paper should acknowledge this and also demonstrate a data-rich example with N > mL+n.
  4. [Section III-C, proof of Proposition 1] In the proof of Proposition 1, the line 'H |_{C(i)} = H |_C' is confusing because C(i) was defined as a set of indices inside I(i), not as a set of rows of H; the notation should be clarified.
  5. [Section V-A, group-LASSO problem (33)] Problem (33) is introduced but no theorem is proved for it; the footnote claiming that 'all results remain valid' needs to be made precise for the group-LASSO formulation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the recovery theorems are proven from stated rank conditions via an externally established fundamental lemma; cited self-work is supporting, not definitional.

full rationale

The paper's derivation chain is self-contained with respect to its assumptions. Lemma 1 imports [17, Corollary 21], a prior peer-reviewed theorem whose stated assumptions (generalized persistency of excitation) do not include the recovery result; this is independent evidence, not an ansatz smuggled in by citation. Theorems 1 and 2 prove sufficiency of Conditions 1 and 2 directly from rank and nullspace arguments, not by assuming the conclusion. The convex recovery guarantees (Theorems 3 and 4) are derived from explicit inequalities (24) and Condition 3 with ||T||_1 < 1; these are sufficient conditions on the offline Hankel matrix and the unknown attack set, not fitted parameters or renamed outputs. The numerical experiments test the algorithms on simulated data and do not fit constants to force success. There is an important correctness caveat: when H has a nontrivial null vector, condition (24) cannot hold and Theorem 4's condition cannot be satisfied, so the l1 guarantees may be vacuous in overdetermined data regimes; however, that is a validity and scope issue rather than circularity. Self-citations to [13] and [17] are load-bearing only as standard lemmas whose proofs are independent of this paper's target claims.

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

The framework rests on the standard behavioral data-driven assumption (generalized persistency of excitation) plus new critical-row-set rank conditions that are hard to verify and depend on the unknown attack set. No new physical entities are introduced and no parameter fitting is used.

free parameters (1)
  • k (upper bound on number of attacked entries or channels)
    Theorems 1-4 and the noisy bi-level procedure assume a known a priori bound k on the adversary's power; in the experiments k=1 is used. This is a user-chosen value, not fitted to data.
assumptions (5)
  • domain assumption Offline trajectory w_d is attack-free and satisfies the generalized persistency of excitation condition rank H_L(w_d) = m(B)L + n(B) (Lemma 1, [17, Corollary 21]).
    Section III states 'We assume that w_d is attack-free. Otherwise, it is not possible to recover the trajectory.' Lemma 1 is the bridge that lets the paper replace the unknown system B|L with the image of the Hankel matrix.
  • domain assumption The system is discrete-time linear time-invariant, and the trajectory segment length L satisfies L >= ell(B), the system lag.
    Lemma 1 and the behavioral formulation require an LTI system and L at least the lag; the paper restricts to LTI in Section II and uses L=3 in the simulations.
  • ad hoc to paper Condition 1 or Condition 2 on the minimum critical row set of the Hankel matrix holds.
    Theorems 1 and 2 assume |S*(H)| >= 2k+1 (entries) or |S~*(H)| >= 2k+1 (channels). These conditions are introduced in this paper; the paper proves they are tight but does not provide an efficient verification method.
  • ad hoc to paper Condition 3 (rank H_B = rank H) and ||T||_1 < 1 hold for the l1 relaxation.
    Theorems 3 and 4 rely on these. H_B is defined via the unknown attack index set C, so the condition is not verifiable in practice; it is essentially an identifiability assumption that the attack does not destroy the row space.
  • domain assumption The set of compromised channels is constant over time in the channel-attacked scenario.
    Section IV states 'we assume that the set of compromised channels remains constant over time, as widely adopted in the literature'.

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Cite this review

Pith. "Pith review of Secure Data Reconstruction: A Direct Data-Driven Approach." pith.science (2026). https://pith.science/paper/TNCV2L4C

@misc{pith2026250200436,
  author       = {Pith},
  title        = {Pith review of: Secure Data Reconstruction: A Direct Data-Driven Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TNCV2L4C}},
  note         = {Machine review of arXiv:2502.00436}
}
abstract

This paper addresses the problem of secure data reconstruction for unknown systems, where data collected from the system are susceptible to malicious manipulation. We aim to recover the real trajectory without prior knowledge of the system model. To achieve this, a behavioral language is used to represent the system, describing it using input/output trajectories instead of state-space models. We consider two attack scenarios. In the first scenario, up to $k$ entries of the collected data are malicious. On the other hand, the second scenario assumes that at most $k$ channels from sensors or actuators can be compromised, implying that any data collected from these channels might be falsified. For both scenarios, we formulate the trajectory recovery problem as an optimization problem and introduce sufficient conditions to ensure successful recovery of the true data. Since finding exact solutions to these problems can be computationally inefficient, we further approximate them using an $\ell_1$-norm and group Least Absolute Shrinkage and Selection Operator (LASSO). We demonstrate that under certain conditions, these approximation problems also find the true trajectory while maintaining low computation complexity. Finally, we extend the proposed algorithms to noisy data. By reconstructing the secure trajectory, this work serves as a safeguard mechanism for subsequent data-driven control methods.

Figures

Figures reproduced from arXiv: 2502.00436 by the authors.

Figure 1
Figure 1. Three interconnected mass spring damper model. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Trajectories under the entry-attacked model by us [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Trajectories under the channel-attacked model by [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Performance comparison among different algorithms [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The average error of each channel [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Trajectories under the channel-attacked model in the quadruple-tank system. conditions are presented to guarantee successful data recon￾struction. Moreover, we develop approximation solutions to mitigate computation costs and handle noisy data. We further demonstrate t…

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Forward citations

Cited by 1 Pith paper

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