Partitioning and Observability in Linear Systems via Submodular Optimization

Ahmad F. Taha; Mohamad H. Kazma

REVIEW 2 major objections 4 minor 47 references

Partitioning and Observability in Linear Systems via Submodular Optimization

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Observability-driven partitioning of linear systems is a submodular maximization problem, solvable with a $(1-1/e)$ guarantee.

desk verdict The new submodular partitioning formulation is worth engaging, but Theorem 2's logdet bound is false as stated and the paper needs major revision before it is citable. read the letter →

arxiv 2505.16169 v3 pith:ECNMMOU6 submitted 2025-05-22 eess.SY cs.SY

Mohamad H. Kazma , Ahmad F. Taha This is my paper · ORCID

classification eess.SYcs.SY MSC 93B0793C0590C27

keywords observabilitynetworkpartitioningsubmodularmaximizationpartitionmatroidsensorplacementlogdetmetricmultilinearextensionlineartime-invariantsystems

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 make observability-driven network partitioning tractable for large linear systems. It claims that splitting the measurable states into $\kappa$ disjoint subsystems so as to maximize the sum of the subsystems' observability metrics can be rewritten as a submodular set-function maximization problem under a partition matroid, which turns the otherwise combinatorial problem into one that greedy algorithms can solve with a $(1-1/e)$ approximation guarantee. The same reformulation then lets sensor placement be solved over the partitioned network, and the paper derives bounds relating the observability of the whole network to the sum of subsystem observabilities. If the claims hold, partitioning offers a scalable route to decentralized state estimation for systems too large for centralized sensor placement.

What carries the argument

The machinery is the reduction of partitioning to submodular maximization under a partition matroid, carried by two objects. First, the subsystem observability Gramian $W_{S_i}$ is a sum of per-state Gramian contributions, making it a modular set function; sums of submodular functions remain submodular, so the logdet and rank metrics over $\kappa$ subsystems inherit submodularity and monotonicity. Second, the multilinear extension of the objective allows the continuous greedy algorithm to operate on fractional assignments, with pipage rounding (or independent randomized rounding for the simple partition matroid) recovering a discrete partition while preserving the $(1-1/e)$ approximation guarantee. The bound that connects local and global observability is the statement that $\log\det(\sum_i W_{S_i}(R_i)) \ge \sum_i \log\det(W_{S_i}(R_i))$, which the paper derives from the Minkowski determinant inequality together with Jensen's inequality.

What would settle it

Evaluate the claimed bound directly on two positive definite matrices, say $W_1=W_2=\mathrm{diag}(10,10)$. Then $\log\det(W_1+W_2)=\log(20\cdot20)\approx 5.99$ while $\log\det(W_1)+\log\det(W_2)=2\log(10\cdot10)\approx 9.21$, so the inequality stated in Theorem 2 fails for this pair; a defender of the theorem would need to show that subsystem Gramians cannot realize such a configuration.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the partition problem and the sensor-placement problem over partitions are both submodular maximization problems. The measurable-state set $V$ is copied once per subsystem to form the ground set $X=\mathcal{C}\times V$, and the constraint $|S\cap(\mathcal{C}\times\{v\})|\le 1$ for each state $v$ says each state is claimed by at most one subsystem. The objective $f(S)=\sum_{i\in\mathcal{C}} f_i(\{v\in V:(i,v)\in S\})$ sums the observability measures of the subsystems, each built from the Gramian $W_{S_i}=\sum_{v\in S_i}\sum_{k=0}^{N-1}(A^k)^\top c_v^\top c_v A^k$. Because this Gramian is modular as a set function, the trace metric is modular while the rank and logdet metrics are monotone submodular, so continuous greedy with the multilinear extension gives a $(1-1/e)$ guarantee for the partition. For sensor placement, the paper's Theorem 2 states that for the trace and logdet metrics, the global observability of the partitioned network is at least the sum of the subsystem observabilities under the same sensor set, and the numerical experiments find that the optimal sensor configuration for the partitioned network coincides with the unpartitioned optimum while taking less computation time.

Load-bearing premise

The load-bearing premise is the determinantal inequality used in Theorem 2, namely that $\log\det(\sum_i W_i)\ge\sum_i\log\det(W_i)$ for the positive semidefinite subsystem Gramians; the proof derives this from Minkowski's determinant inequality and Jensen's inequality, and the bound collapses if that derivation is not valid.

Editorial extensions

If this is right

The partition problem P1 becomes solvable in polynomial time: continuous greedy under the partition matroid achieves the $(1-1/e)$ guarantee, and pipage rounding returns a feasible partition.
Sensor placement over the partitioned network inherits the same submodularity, so greedy sensor selection with per-subsystem quotas carries a provable approximation factor.
For the trace and logdet metrics, the global observability measure evaluated at any sensor configuration is at least the sum of the subsystem observability measures at that same configuration; for the rank metric the inequality reverses.
On the tested combustion networks, the optimal sensor configuration obtained after partitioning matches the unpartitioned optimum, so partitioning need not sacrifice observability while cutting sensor-placement computation time.
The results provide a principled way to choose the number of partitions $\kappa$: the optimal $\kappa$ balances subsystem observability against inter-subsystem connections, with $\kappa=5$ best on the GRI30 network.

Reading between the lines

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

By duality, the same submodular machinery should reformulate actuator placement under controllability Gramians; the paper notes this in passing but does not develop it, so a natural extension is to run the identical pipeline on the dual system.
A subtlety left implicit is that the partition matroid in P2 only prevents a state from being claimed twice and does not by itself force every state to be claimed, so degenerate cases with zero marginal gains could return a partial partition unless full coverage is separately enforced.
The bound in Theorem 2, if valid, implies that the gap between global and local sensor objectives grows with the disparity in subsystem Gramian scales; this is testable by running P3 on networks with deliberately heterogeneous subsystem sizes.
The continuous greedy algorithm for the simple partition matroid can use independent randomized rounding instead of pipage rounding, so the method is amenable to parallelization across subsystems for very large networks.

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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

Here's the short version: the paper has a genuinely new problem formulation — partitioning an LTI state space into observable subsystems via submodular welfare maximization over duplicated states — and the submodularity claims for trace, rank, and logdet under a partition are mostly standard but correctly assembled. The continuous greedy treatment is textbook and cited properly. But the paper's headline comparison result, Theorem 2, is false as stated. The logdet lower bound logdet(Σ W_i) ≥ Σ logdet(W_i) fails even for two equal PSD blocks: take W1=W2=10·I_2; left side is 2 ln 20 ≈ 5.99, right side is 4 ln 10 ≈ 9.21. The proof's Jensen step uses weights 1/α_i whose sum is κ, not 1, and the direction is reversed for a concave function. This is not a minor gap; the bound is advertised in the abstract, Section V, and conclusions, and the experiments in Section VII purport to validate it. It also infects the interpretation of Tab. III, where the gap between f(∪ R_i) and Σ f(R_i) is taken as evidence for the bound. The numerical results are suggestive but cannot repair a false theorem.

There is a second, smaller flaw in Lemma 1: P2's partition matroid only prevents a state from being assigned twice; it does not require full coverage of V. The proof asserts 'since v∈V is allocated to at least one partition' without justification. The bijection claim is therefore not correct, though for monotone f the optimal values may still coincide because any optimal solution will assign every state somewhere. This should be restated as an optimality-equivalence, not a feasible-space bijection, or the constraint needs to be tightened (e.g., add a cardinality lower bound). The rest of the machinery — modularity of W_Si, submodularity of logdet/rank via interpolation, the (1−1/e) guarantee for P2/P3 — holds up structurally, as far as I can tell.

Who is this for? Researchers working on decentralized state estimation and sensor placement for large LTI systems will find the formulation attractive, but they should not rely on Theorem 2 until it is corrected or replaced. The paper deserves a serious referee — the core idea is worth engaging with — but it needs major revision before publication. I'd send it to review rather than desk-reject, with the expectation that the authors fix the bound or explicitly drop it.

Referee Report

2 major / 4 minor

Summary. The paper studies the problem of partitioning a linear time-invariant system into κ disjoint subsystems while maximizing observability-based metrics (trace, rank, logdet) of the subsystem dynamics, and then solving sensor placement over the partitioned system. The authors formulate the partitioning problem as a submodular maximization under a partition matroid (P2), claim equivalence with the original partition problem (P1), prove submodularity of the observability measures, and derive a comparison bound between the global observability of the partitioned system and the sum of subsystem observabilities. Numerical experiments on two combustion reaction networks are used to validate the approach and the theoretical bounds.

Significance. The problem is practically relevant, and the use of submodular optimization with matroid constraints is a suitable framework; the numerical studies are clearly described and show computational savings for sensor placement on partitioned networks. However, the validity of the paper's central theoretical claims is not established. In particular, Theorem 2, which is presented as a key contribution and is cited as being validated by the experiments, is false, and Lemma 1's asserted equivalence between P1 and P2 is not correctly proven. These are load-bearing issues because the abstract, the concluding remarks, and the experimental discussion all rely on these results. The submodular reformulation may still be salvageable, but the current manuscript's headline claims are unsupported.

major comments (2)

[Section V-B, Eq. (12)] The lower bound in Theorem 2 for the logdet metric is false as stated. In the proof, after defining α_i = r_i/r, Jensen's inequality is applied with coefficients 1/α_i whose sum is κ, not 1, so that step is invalid. More decisively, the claimed inequality logdet(Σ_i W_i) ≥ Σ_i logdet(W_i) fails for PSD matrices: for W_1 = W_2 = 10 I_2, the left-hand side equals 2 ln 20 ≈ 5.99 while the right-hand side equals 4 ln 10 ≈ 9.21. This false theorem is invoked in the abstract, in Section VII (Table III), and in the concluding remarks, so the central claim that the partitioned system's observability is bounded below by the sum of subsystem observabilities is unsupported.
[Section III, Lemma 1] The claimed equivalence between P1 and P2 is not correct as proven. The partition matroid constraint (4b) only enforces |S ∩ (C × {v})| ≤ 1, i.e., each state is assigned to at most one subsystem; it does not require every state to be assigned to some subsystem. In the proof, the sentence 'since v ∈ V is allocated to at least one partition then ∪_i S_i = V' is asserted without justification. Consequently, P2 is a relaxation of P1, and the feasible solution spaces are not bijective. While monotonicity of the objective may imply that an optimal P2 solution covers all of V, the lemma as stated is false and must be reformulated or proved with an explicit coverage argument.

minor comments (4)

[Section V-B] The notation f(∪_{i∈C} W_{S_i}(R_i)) uses a union symbol for matrices; the intended operation is the sum, as used in the proof and in Table I.
[Section V-B, Corollary 4] The submodularity proof for logdet is deferred with 'For brevity, we do not show the matrix derivative'; since Corollary 4 underlies the greedy guarantees for P3, the omitted derivative should be supplied or a complete reference given.
[Section VII-B] The phrase 'Section IV .B' is malformed; it should read 'Section IV-B' or 'Section IV.B'.
[Section VI] The complexity statements O(|X|^7) and O(|X|^2) are quoted without specifying the exact algorithm variant or the oracle cost model; please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain uses external submodular-optimization results and standard linear algebra, with no self-citation or fit-renamed-as-prediction; the main theoretical weaknesses are mathematical errors, not circular reasoning.

full rationale

The paper's central claims are not circular. The submodular formulation P2, the continuous-greedy (1-1/e) guarantee, and the submodularity lemmas are supported by in-paper arguments combined with external, independently established results ([28], [29], [32], [33]), not by the authors' own prior uniqueness theorems or fitted parameters. The partitioned-versus-unpartitioned sensor-placement comparisons in Section VII are computed independently (including a brute-force check of the unpartitioned optimum) and are not obtained by renaming an input as a prediction. The Gramian decomposition in Proposition 1 and Corollary 1 is a valid algebraic identity following from disjoint coverage of the state set, not a self-definitional reduction. What the reader's take identifies are genuine mathematical flaws: Lemma 1's equivalence claim is weakened because the P2 matroid constraint only enforces 'at most one' assignment and does not itself force full coverage of V, and Theorem 2's logdet lower bound rests on an invalid Jensen step with non-unit weights and is false for equal PSD matrices. These are correctness risks, not circularity patterns: no quantity is fitted then predicted, no result is imported solely from a self-citation, and no conclusion is identical to its premises by construction. Accordingly, the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The derivation relies on standard linear algebra and submodularity facts, plus one false inequality in Theorem 2. No new physical entities are introduced. κ and r are user-specified design variables rather than fitted constants, and the numerical settings N and Δt are experiment choices that do not enter the theoretical claims.

free parameters (4)

Number of partitions κ = 3-6 in GRI30 case study
User-specified design variable; range chosen from modularity analysis, not fitted.
Number of sensors r = 10, 16, 23 in case study
User-specified design variable for sensor placement.
Observation horizon N = 1000
Chosen for computing finite-horizon observability Gramians.
Discretization step Δt = 1e-12
Integration step for linearizing the combustion networks.

assumptions (6)

standard math Finite-horizon observability Gramian decomposes as a sum of per-state terms W_v = Σ_k (A^k)^T c_v^T c_v A^k.
Stated in Proposition 1 and used in every Gramian computation.
standard math A nonnegative weighted sum of submodular functions is submodular.
Lemma 2, cited from the literature; used to combine subsystem measures.
standard math rank and logdet of observability Gramians are monotone submodular set functions under addition of measurement rows.
Known in sensor selection; used in Corollaries 2 and 4.
domain assumption Every state in V can be sensed by a to-be-placed sensor, and C rows correspond one-to-one with measurable states.
Assumed in Section II-B; needed for per-state decomposition.
domain assumption Every state in V is allocated to exactly one subsystem in the P2 reformulation.
Lemma 1 assumes full coverage, but the partition matroid only enforces at most one assignment.
ad hoc to paper Minkowski determinant inequality and Jensen's inequality combine to give logdet(Σ_i W_i) ≥ Σ_i logdet(W_i).
This premise is false; Jensen is applied with weights 1/α_i that do not sum to 1.

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Cite this review

Pith. "Pith review of Partitioning and Observability in Linear Systems via Submodular Optimization." pith.science (2026). https://pith.science/paper/ECNMMOU6

@misc{pith2026250516169,
  author       = {Pith},
  title        = {Pith review of: Partitioning and Observability in Linear Systems via Submodular Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECNMMOU6}},
  note         = {Machine review of arXiv:2505.16169}
}

read the original abstract

Network partitioning has gained recent attention as a pathway to enable decentralized operation and control in large-scale systems. This paper addresses the interplay between partitioning, observability, and sensor placement (SP) in dynamic networks. The problem, being computationally intractable at scale, is a largely unexplored, open problem in the literature. To that end, the paper's objective is designing scalable partitioning of linear systems while maximizing observability metrics of the subsystems. We show that the partitioning problem can be posed as a submodular maximization problem -- and the SP problem can subsequently be solved over the partitioned network. Consequently, theoretical bounds are derived to compare observability metrics of the original network with those of the resulting partitions, highlighting the impact of partitioning on system observability. Case studies on networks of varying sizes corroborate the derived theoretical bounds.

Figures

Figures reproduced from arXiv: 2505.16169 by the authors.

**Figure 1.** (a) A system of measured internal state, v ∈ V (measurable space), depicting interactions of a dynamical system and (b) the subsequent subsystems Si ⊆ V. The nodes represent the system states and edges represent the internal state connections. The colored boxes represent the nodes that belong to a particular subsystem. The interactions between the subsystems remain after system partitioning. result, one potential g… view at source ↗

**Figure 2.** Matroid Constraints: For any (a) submodular maximization problem, a matroid constraint (b) satisfies: (i) the null property, (ii) the heredity property and (iii) augmentation property. Two common matroid constraints are the (c) uniform matroid Ic and (d) partition matroid Ip. submodular objective functions are often constrained on an admissible subset S ⊆ V of the ground set. Constraints that arise in the context of… view at source ↗

**Figure 3.** Overview of the system partitioning problem under P2 (Sections II–IV), followed by SP under P3 (Sections V). The continuous greedy algorithm can be utilized to solve P2 and P3 (Section VI) for an LTI system such as a linearized combustion reaction network (Section VII). where k ∈ N refers to the discrete-time index, such that vector x[k] ∈ R nx represents the dynamic states, vector u[k] ∈ R nu represents the inputs … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: (a) Disjoint selection of state v from ground set V. (b) Disjoint selection from ground set X = C × V by duplicating each element v ∈ V for each subsystem i ∈ C; selection under a partition matroid. process using a partition matroid with κi = 1 for all i ∈ C, we ensure…

**Figure 5.** Figure 5: (a) Mapping of state subsets Si to outputs ySi [k]; each vi corresponds to a row cv of C. (b) Composition of the full observability Gramian Wo from subsystem Gramians WSi and the corresponding matrix structure in each summation term of the subsystem Gramians. The syste…

**Figure 6.** Figure 6: The framework for submodular partitioning and sensor placement in LTI systems. The left column partitions the measurable statespace by solving P2 while quantifying observability based on κ subsystems. The right column represents the resulting partitioned system, whe…

**Figure 7.** Figure 7: (N1) Partitioning of an H2O2 combustion reaction network. Sensor locations on the (a) unpartitioned system and (b) partitioned system that is clustered by color. Sensor nodes can be identified by observing the shaded nodes within each cluster and the colored nodes with…

**Figure 8.** Figure 8: Partition modularity Q of the GRI30 combustion network obtained from spectral clustering for varying κ. The selected range for κ is shown in red. For each κ, we compute the first κ eigenvectors corresponding to the smallest non-zero eigenvalues of L¯ and use them to c…

**Figure 9.** Figure 9: (N2) Partitioning of a GRI30 combustion reaction network. Sensor locations for the (a) unpartitioned system and partitioned systems with (b) κ = 3, (c) κ = 4, (d) κ = 5, and (e) κ = 6 are shown. Sensor nodes can be identified by observing the shaded nodes within each s…

**Figure 10.** Figure 10: Modularity Q of the optimal partitions and the ratio of the observability measure f(S) for subsystems obtained by solving P2 versus those obtained via spectral clustering for the GRI30 network. function still contributes to maximizing global observability. The partiti…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Detection of Covert Cyber-Attacks in Interconnected Systems: A Distributed Model-Based Approach,

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Complex networks: Structure and dynamics,

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On Submodularity and Controllability in Complex Dynamical Networks,

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Performance guarantees for greedy maximization of non-submodular controllability metrics,

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Sensor Assignment Algorithms to Improve Observability while Tracking Targets,

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Sensor Selection for Dynamics-Driven User-Interface Design,

A. P. Vinod, A. J. Thorpe, P. A. Olaniyi, T. H. Summers, and M. M. K. Oishi, “Sensor Selection for Dynamics-Driven User-Interface Design,” IEEE Transactions on Control Systems Technology, vol. 30, no. 1, pp. 71–84, 2022

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