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arxiv: 2604.13862 · v1 · submitted 2026-04-15 · 📡 eess.SY · cs.SY

Data-Driven Reachability Analysis Using Matrix Perturbation Theory

Peng Xie , Abdulla Fawzy , Zhen Zhang , Amr Alanwar This is my paper

Pith reviewed 2026-05-10 12:41 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven reachability analysismatrix zonotopeperturbation theoryconstrained zonotopereachable setsuncertain dynamical systemscoefficient space approximation
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The pith

A coefficient-space approximation for constrained matrix zonotopes enables faster and less conservative data-driven reachability analysis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a matrix zonotope perturbation framework that uses matrix perturbation theory to bound how noise distorts the dynamics of sets of models. It derives Cai-Zhang bounds first for matrix zonotopes and then extends them to constrained matrix zonotopes. Motivated by the computational cost of propagating the constrained versions, the authors replace the constrained coefficient space with an over-approximating unconstrained zonotope. This change turns expensive constrained products into simple unconstrained multiplications during reachable-set updates. A sympathetic reader would care because reachability analysis is central to safety verification of uncertain systems learned from data, yet existing approaches are either slow or overly conservative.

Core claim

By over-approximating the constrained coefficient space of a constrained matrix zonotope with an unconstrained zonotope, reachable-set propagation reduces to unconstrained matrix-zonotope times zonotope multiplication. This update respects the perturbation bounds while being substantially faster than full constrained-matrix-zonotope constrained-zonotope operations and less conservative than existing matrix-zonotope methods.

What carries the argument

The coefficient-space approximation, which replaces the constrained coefficient space of the CMZ by an unconstrained zonotope so that reachable-set updates use only MZ-zonotope multiplication instead of CMZ-CZ products.

If this is right

  • Reachable sets for data-driven models with noise become computable in real time.
  • Safety verification of uncertain dynamical systems becomes more practical without sacrificing bound interpretability.
  • Reachable sets remain tighter than those produced by prior matrix-zonotope techniques.
  • The same perturbation analysis extends naturally from single models to entire sets of models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same approximation step could be tested on other constrained set representations used in reachability analysis.
  • Online safety monitoring in robotics or autonomous vehicles could adopt this faster propagation for live verification.
  • High-dimensional numerical studies would reveal how the extra conservatism scales with system size.
  • The framework could combine with data-driven set identification methods to close the loop from measurements to verified reachable sets.

Load-bearing premise

The over-approximation of the constrained coefficient space by an unconstrained zonotope does not introduce unacceptable extra conservatism or invalidate the perturbation bounds for the intended applications.

What would settle it

Execute the method on a standard benchmark system, compute the reachable-set volume and wall-clock time, and compare directly against both the full CMZ method and existing MZ methods; if the new sets are larger than those from CMZ or slower than claimed, or more conservative than MZ results, the performance advantage does not hold.

Figures

Figures reproduced from arXiv: 2604.13862 by Abdulla Fawzy, Amr Alanwar, Peng Xie, Zhen Zhang.

Figure 1
Figure 1. Figure 1: Noisy state–input data give rise to a set of models for the pair [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reachable-set projections under different state [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: With fixed noise, enlarging state-input data mag [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A comparison between the exact set of possible coefficients [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: right 4000 0.0065 145.73 0.0035 5.281 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of reachable-set projections using matrix zonotope, constrained matrix zonotope, the nullspace matrix [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Piecewise affine system (switching at 𝑥1 = 0). Initial set X0 (black); model-based set R𝑘 (blue); MZ-based set Rˆ 𝑘 (black solid line); NMZ-based set RˆNMZ 𝑘 (red). The NMZ yields a tighter outer approximation. (NMZ), which retains the unconstrained zonotope structure of the MZ yet delivers substantially tighter outer approximations than the CMZ when the number of generators of the reachable set is limited… view at source ↗
read the original abstract

We propose a matrix zonotope perturbation framework that leverages matrix perturbation theory to characterize how noise-induced distortions alter the dynamics within sets of models. The framework derives interpretable Cai-Zhang bounds for matrix zonotopes (MZs) and extends them to constrained matrix zonotopes (CMZs). Motivated by this analysis and the computational burden of CMZ-based reachable-set propagation, we introduce a coefficient-space approximation in which the constrained coefficient space of the CMZ is over-approximated by an unconstrained zonotope. Replacing CMZ-constrained-zonotope (CZ) products with unconstrained MZ-zonotope multiplication yields a simpler and more scalable reachable-set update. Experimental results demonstrate that the proposed method is substantially faster than the standard CMZ approach while producing reachable sets that are less conservative than those obtained with existing MZ-based methods, advancing practical, accurate, and real-time data-driven reachability analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a matrix zonotope perturbation framework that leverages matrix perturbation theory to characterize noise-induced distortions in sets of models for data-driven reachability analysis. It derives interpretable Cai-Zhang bounds for matrix zonotopes (MZs) and extends them to constrained matrix zonotopes (CMZs). Motivated by the computational cost of CMZ propagation, the authors introduce a coefficient-space approximation that over-approximates the constrained coefficient space of a CMZ by an unconstrained zonotope, enabling simpler MZ-zonotope multiplications for reachable-set updates. Experimental results are claimed to show that the method is substantially faster than standard CMZ approaches while producing reachable sets less conservative than those from existing MZ-based methods.

Significance. If the approximation error is rigorously bounded and the experimental claims are supported by detailed, reproducible benchmarks, the work could advance practical data-driven reachability analysis by offering a computationally scalable alternative that retains interpretability via perturbation bounds. The explicit use of Cai-Zhang theory for zonotope-based uncertainty propagation is a positive aspect for applications in control and verification where real-time performance matters.

major comments (2)
  1. [coefficient-space approximation section] The coefficient-space approximation (described after the CMZ extension) replaces constrained-zonotope products with unconstrained MZ-zonotope multiplication by over-approximating the coefficient space. No explicit bound (e.g., Hausdorff distance, volume ratio, or perturbation-error propagation) is provided on the additional conservatism this step introduces relative to exact CMZ or to the MZ baselines. Without such quantification, it is unclear whether the reachable sets remain less conservative than existing MZ methods as asserted in the abstract and conclusion.
  2. [experimental results section] The experimental claims of substantial speed-up and reduced conservatism lack supporting details: no description of benchmark systems, noise models, reachable-set error metrics (volume, Hausdorff distance, or over-approximation ratio), number of Monte-Carlo runs, or statistical significance testing. These omissions make it impossible to assess whether the performance gains are robust or whether the approximation's extra conservatism is negligible in the tested regimes.
minor comments (2)
  1. [preliminaries] Notation for matrix zonotopes, constrained matrix zonotopes, and their generators should be collected in a single table or preliminary section for clarity, as the distinction between coefficient constraints and matrix entries is central to the approximation.
  2. [abstract] The abstract states performance superiority without referencing any specific table or figure; adding a forward reference to the relevant experimental table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive review of our manuscript. We address each of the major comments below and outline the revisions we will make to improve the paper.

read point-by-point responses

  1. Referee: [coefficient-space approximation section] The coefficient-space approximation (described after the CMZ extension) replaces constrained-zonotope products with unconstrained MZ-zonotope multiplication by over-approximating the coefficient space. No explicit bound (e.g., Hausdorff distance, volume ratio, or perturbation-error propagation) is provided on the additional conservatism this step introduces relative to exact CMZ or to the MZ baselines. Without such quantification, it is unclear whether the reachable sets remain less conservative than existing MZ methods as asserted in the abstract and conclusion.

    Authors: We appreciate the referee highlighting this important point regarding the lack of an explicit bound on the conservatism introduced by the coefficient-space approximation. While the approximation ensures that the reachable sets remain sound over-approximations (as it enlarges the coefficient space), we acknowledge that a quantitative bound relative to the exact CMZ or comparison to MZ methods would strengthen the claims. In the revised manuscript, we will add a new subsection providing a bound on the Hausdorff distance between the approximated and exact reachable sets, derived from the properties of the zonotope over-approximation. This will also include a discussion on how the error relates to the perturbation bounds from Cai-Zhang theory, supporting the assertion that the results are less conservative than pure MZ approaches. revision: yes

  2. Referee: [experimental results section] The experimental claims of substantial speed-up and reduced conservatism lack supporting details: no description of benchmark systems, noise models, reachable-set error metrics (volume, Hausdorff distance, or over-approximation ratio), number of Monte-Carlo runs, or statistical significance testing. These omissions make it impossible to assess whether the performance gains are robust or whether the approximation's extra conservatism is negligible in the tested regimes.

    Authors: We agree that the experimental section requires additional details for full reproducibility and to substantiate the claims. In the revised version, we will expand this section significantly. Specifically, we will describe the benchmark systems used (including system dimensions, initial sets, and input constraints), the noise models (e.g., the specific bounded disturbance assumptions), the metrics for evaluating reachable sets (such as volume computation, Hausdorff distance to ground truth where available, and over-approximation ratios), the number of Monte-Carlo simulations performed for validation, and any statistical tests applied. We will also include tables summarizing the results with these metrics and make the implementation code available upon request or via a repository to allow verification of the speed-up and conservatism claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained with explicit approximation trade-off

full rationale

The paper first derives Cai-Zhang perturbation bounds for matrix zonotopes, extends the analysis to constrained matrix zonotopes, and then explicitly introduces a coefficient-space over-approximation (replacing constrained coefficient space with an unconstrained zonotope) as a deliberate computational simplification. This enables faster MZ-style multiplication while the soundness and relative conservatism claims are supported by separate experimental comparisons rather than by construction. No step reduces a prediction or bound to a fitted parameter, self-citation chain, or renamed input; the approximation is presented as an engineering choice whose extra conservatism is accepted in exchange for speed, with no tautological redefinition of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard matrix perturbation theory and zonotope set representations from control theory; the coefficient-space approximation is the primary new modeling choice.

axioms (2)
  • domain assumption Matrix perturbation theory yields interpretable bounds on how additive noise distorts matrix sets
    Invoked to derive Cai-Zhang bounds for matrix zonotopes and their constrained extensions.
  • standard math Zonotope arithmetic and over-approximations preserve soundness for reachability propagation
    Underlying the reachable-set update step after the coefficient-space approximation.

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Alireza Naderi Akhormeh, Amr Hegazy, and Amr Alanwar. 2025. Online Data- Driven Reachability Analysis using Zonotopic Recursive Least Squares.arXiv preprint arXiv:2509.17058(2025)

    work page arXiv 2025

  2. [2]

    Amr Alanwar, Alexander Berndt, Karl Henrik Johansson, and Henrik Sandberg

    work page

  3. [3]

    Funnel Control for Relative Degree One Nonlinear Systems With Input Saturation, in: 2022 European Control Conference (ECC), IEEE, London, United Kingdom

    Data-Driven Set-Based Estimation using Matrix Zonotopes with Set Con- tainment Guarantees. In2022 European Control Conference (ECC). IEEE, 875–881. doi:10.23919/ECC55457.2022.9838494

    work page doi:10.23919/ecc55457.2022.9838494 2022

  4. [4]

    Amr Alanwar, Anne Koch, Frank Allgöwer, and Karl Henrik Johansson. 2023. Data-driven reachability analysis from noisy data.IEEE Trans. Automat. Control 68, 5 (2023), 3054–3069. doi:10.1109/TAC.2023.3257167

    work page doi:10.1109/tac.2023.3257167 2023

  5. [5]

    Matthias Althoff. 2015. An introduction to CORA 2015. InProceedings of the Workshop on Applied Verification for Continuous and Hybrid Systems (ARCH) (EPiC Series in Computing, Vol. 34). EasyChair, 120–151

    work page 2015

  6. [6]

    Krogh, and Olaf Stursberg

    Matthias Althoff, Bruce H. Krogh, and Olaf Stursberg. 2011. Analyzing Reacha- bility of Linear Dynamic Systems with Parametric Uncertainties. InModeling, Design, and Simulation of Systems with Uncertainties, Andreas Rauh and Ekaterina Auer (Eds.). Mathematical Engineering, Vol. 3. Springer, Berlin, Heidelberg, 69–94. doi:10.1007/978-3-642-15956-5_4

    work page doi:10.1007/978-3-642-15956-5_4 2011

  7. [7]

    Distributed event-triggered control for global consensus of multi-agent systems with input saturation,

    Trevor J. Bird, Herschel C. Pangborn, Neera Jain, and Justin P. Koeln. 2023. Hybrid zonotopes: A new set representation for reachability analysis of mixed logical dynamical systems.Automatica154 (2023), 111107. doi:10.1016/j.automatica. 2023.111107

    work page doi:10.1016/j.automatica 2023

  8. [8]

    2004.Convex Optimization

    Stephen Boyd and Lieven Vandenberghe. 2004.Convex Optimization. Cambridge University Press

    work page 2004

  9. [9]

    Tony Cai and Anru Zhang

    T. Tony Cai and Anru Zhang. 2018. Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics.The Annals of Statistics 46, 1 (2018), 60–89. doi:10.1214/17-AOS1541

    work page doi:10.1214/17-aos1541 2018

  10. [10]

    Charnes and W

    A. Charnes and W. W. Cooper. 1962. Programming with linear fractional func- tionals.Naval Research Logistics Quarterly9, 3-4 (1962), 181–186. doi:10.1002/ nav.3800090303

    work page 1962

  11. [11]

    Chandler Davis and William Morton Kahan. 1970. The rotation of eigenvectors by a perturbation. III.SIAM J. Numer. Anal.7, 1 (1970), 1–46. doi:10.1137/0707001

    work page doi:10.1137/0707001 1970

  12. [12]

    Mahsa Farjadnia, Angela Fontan, Amr Alanwar, Marco Molinari, and Karl Henrik Johansson. 2024. Robust data-driven tube-based zonotopic predictive control with closed-loop guarantees. In2024 IEEE 63rd Conference on Decision and Control (CDC). IEEE, 6837–6843. doi:10.1109/CDC56724.2024.10886128

    work page doi:10.1109/cdc56724.2024.10886128 2024

  13. [13]

    Antoine Girard. 2005. Reachability of Uncertain Linear Systems Using Zonotopes. InHybrid Systems: Computation and Control (HSCC) (LNCS, Vol. 3414). Springer, 291–305. doi:10.1007/978-3-540-31954-2_19

    work page doi:10.1007/978-3-540-31954-2_19 2005

  14. [14]

    Antoine Girard and Colas Le Guernic. 2008. Zonotope/hyperplane intersection for hybrid systems reachability analysis. InHybrid Systems: Computation and Control (HSCC) (LNCS, Vol. 4981). Springer, 215–228. doi:10.1007/978-3-540-78929-1_16

    work page doi:10.1007/978-3-540-78929-1_16 2008

  15. [15]

    Niklas Kochdumper and Matthias Althoff. 2023. Constrained polynomial zono- topes.Acta Informatica60, 3 (2023), 279–316. doi:10.1007/s00236-023-00437-5

    work page doi:10.1007/s00236-023-00437-5 2023

  16. [16]

    Anna-Kathrin Kopetzki, Bastian Schürmann, and Matthias Althoff. 2017. Methods for order reduction of zonotopes. In2017 IEEE 56th Annual Conference on Decision and Control (CDC). IEEE, 5626–5633. doi:10.1109/CDC.2017.8264508

    work page doi:10.1109/cdc.2017.8264508 2017

  17. [17]

    Alan Laub. 2012. The Moore–Penrose Pseudoinverse (Math 33A). UCLA Notes. https://www.math.ucla.edu/~laub/33a.2.12s/mppseudoinverse.pdf Data-Driven Reachability Analysis Using Matrix Perturbation Theory HSCC ’26, May 11–14, 2026, Saint Malo, France

    work page 2012

  18. [18]

    Lech Maligranda. 2006. Simple norm inequalities.The American Mathematical Monthly113, 3 (2006), 256–260. doi:10.1080/00029890.2006.11920303

    work page doi:10.1080/00029890.2006.11920303 2006

  19. [19]

    Patrick E. O’Neil. 1971. Hyperplane cuts of an 𝑛-cube.Discrete Mathematics1, 2 (1971), 193–195. doi:10.1016/0012-365X(71)90025-2

    work page doi:10.1016/0012-365x(71)90025-2 1971

  20. [20]

    Sean O’Rourke, Van Vu, and Ke Wang. 2024. Matrices with Gaussian noise: Optimal estimates for singular subspace perturbation.IEEE Transactions on Information Theory70, 3 (2024), 1978–2002. doi:10.1109/TIT.2023.3331010

    work page doi:10.1109/tit.2023.3331010 2024

  21. [21]

    Mohammed Adib Oumer, Amr Alanwar, and Majid Zamani. 2025. Data-Driven Safety Verification using Barrier Certificates and Matrix Zonotopes. In2025 IEEE 64th Conference on Decision and Control (CDC). IEEE, 3913–3918. doi:10.1109/ CDC57313.2025.11312081

    work page arXiv 2025

  22. [22]

    Kaare Brandt Petersen and Michael Syskind Pedersen. 2012. The Matrix Cook- book. https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf

    work page 2012

  23. [23]

    Vignesh Raghuraman and Justin P. Koeln. 2022. Set operations and order reduc- tions for constrained zonotopes.Automatica139 (2022), 110204. doi:10.1016/j. automatica.2022.110204

    work page doi:10.1016/j 2022

  24. [24]

    Rego, Davide M

    Brenner S. Rego, Davide M. Raimondo, and Guilherme V. Raffo. 2025. Line zonotopes: A tool for state estimation and fault diagnosis of unbounded and descriptor systems.Automatica179 (2025), 112380. doi:10.1016/j.automatica.2025. 112380

    work page doi:10.1016/j.automatica.2025 2025

  25. [25]

    Francisco Rego and Daniel Silvestre. 2025. A novel and efficient order reduction for both constrained convex generators and constrained zonotopes.IEEE Trans. Automat. Control70, 6 (2025), 4016–4023. doi:10.1109/TAC.2024.3525388

    work page doi:10.1109/tac.2024.3525388 2025

  26. [26]

    Alessio Russo and Alexandre Proutiere. 2023. Tube-Based Zonotopic Data-Driven Predictive Control. In2023 American Control Conference (ACC). IEEE, 3845–3851. doi:10.23919/acc55779.2023.10156056

    work page doi:10.23919/acc55779.2023.10156056 2023

  27. [27]

    2013.Convex Bodies: The Brunn–Minkowski Theory(2 ed.)

    Rolf Schneider. 2013.Convex Bodies: The Brunn–Minkowski Theory(2 ed.). Cam- bridge University Press

    work page 2013

  28. [28]

    Scott, Davide M

    Joseph K. Scott, Davide M. Raimondo, Giuseppe R. Marseglia, and Richard D. Braatz. 2016. Constrained Zonotopes: A New Tool for Set-Based Estimation and Fault Detection.Automatica69 (2016), 126–136. doi:10.1016/j.automatica.2016.02. 036

    work page doi:10.1016/j.automatica.2016.02 2016

  29. [29]

    Joel A. Tropp. 2015. An introduction to matrix concentration inequalities.Foun- dations and Trends®in Machine Learning8, 1-2 (2015), 1–230. doi:10.1561/ 2200000048

    work page 2015

  30. [30]

    Per-Åke Wedin. 1972. Perturbation bounds in connection with singular value decomposition.BIT Numerical Mathematics12, 1 (1972), 99–111. doi:10.1007/ BF01932678

    work page 1972

  31. [31]

    Hermann Weyl. 1912. Das asymptotische Verteilungsgesetz der Eigenwerte lin- earer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung).Math. Ann.71, 4 (1912), 441–479. doi:10.1007/BF01456804

    work page doi:10.1007/bf01456804 1912

  32. [32]

    Peng Xie, Johannes Betz, Davide M Raimondo, and Amr Alanwar. 2025. Data- Driven Reachability Analysis for Piecewise Affine Systems. In2025 IEEE 64th Con- ference on Decision and Control (CDC). IEEE, 1356–1363. doi:10.1109/CDC57313. 2025.11312136

    work page doi:10.1109/cdc57313 2025

  33. [33]

    Zhen Zhang, M Umar B Niazi, Michelle S Chong, Karl H Johansson, and Amr Alanwar. 2025. Data-Driven Nonconvex Reachability Analysis Using Exact Multiplication. In2025 IEEE 64th Conference on Decision and Control (CDC). IEEE, 4882–4889. doi:10.1109/CDC57313.2025.11312292

    work page doi:10.1109/cdc57313.2025.11312292 2025

  34. [34]

    Günter M. Ziegler. 1995.Lectures on Polytopes(1 ed.). Graduate Texts in Mathe- matics, Vol. 152. Springer, New York

    work page 1995