PCA-DDReach: Efficient Statistical Reachability Analysis of Stochastic Dynamical Systems via Principal Component Analysis

Jyotirmoy Deshmukh; Lars Lindemann; Navid Hashemi

REVIEW 3 major objections 6 minor 28 references

PCA-DDReach: Efficient Statistical Reachability Analysis of Stochastic Dynamical Systems via Principal Component Analysis

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read PCA-aligned error boxes and per-segment surrogate models make statistically guaranteed reachable sets smaller, distribution-shift-safe, and scalable to thousands of steps.

desk verdict Solid extension of the conformal reachability line with a real scalability payoff; the unverified TV bound in the shift experiment is the main obstacle. read the letter →

arxiv 2505.14935 v1 pith:CIPJM6WL submitted 2025-05-20 cs.RO

Navid Hashemi , Lars Lindemann , Jyotirmoy Deshmukh This is my paper · ORCID

classification cs.RO

keywords reachablesetestimationconformalinferenceprincipalcomponentanalysisdistributionshiftstochasticdynamicalsystemsstatisticalverificationneuralnetworkreachabilitystarsets

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

The paper addresses data-driven reachability analysis: given only sampled trajectories from a stochastic black-box system, compute a set that contains any deployment trajectory with probability at least $\delta$, even when deployment data differ from training data. It claims that two changes to a prior conformal-inference method make this practical: train one small surrogate model per trajectory segment instead of one monolithic model, and build the inflating error box in a coordinate system aligned with the principal components of the prediction errors rather than along the original state axes. The PCA alignment centers the box at the mean prediction error and orients it along the directions of largest variation, so the $\delta$-confident flowpipe is tighter, while the segmented training removes cumulative error growth and allows exact-star reachability analysis on smaller networks. The paper validates the method on a 12-dimensional quadcopter and a 27-dimensional hybrid powertrain with a bounded distribution shift.

What carries the argument

The load-bearing object is the PCA-calibrated residual $\rho = \max_j(|r_j|/\omega_j)$. The eigenvectors $V$ and the scales $\omega_j$ are computed from the training trajectories' prediction errors; projecting onto $V$ rotates the coordinate system so the error cloud is axis-aligned, and dividing by $\omega_j$ normalizes each axis. Robust conformal inference on a separate calibration set returns the scalar $\rho^*_{\delta,\tau}$, and the identity $\rho < \rho^*_{\delta,\tau} \Leftrightarrow \bigwedge_j (|r_j| < \rho^*_{\delta,\tau} \omega_j)$ turns that one-dimensional quantile into a multidimensional parallelotope. Concatenating the per-segment parallelotopes and Minkowski-adding them to the surrogate flowpipe produces the $\delta$-confident flowpipe; the same equivalence is what lets the paper state Proposition 5 as a finite conjunction over segments.

What would settle it

Run the powertrain experiment with process-noise covariance increased by 25% instead of 20%, keep $\tau = 0.04$ and $\delta = 0.95$, simulate many deployment trajectories, and count the fraction that fall outside the computed flowpipe; if that empirical miss rate exceeds 5%, the stated distribution-shift guarantee is violated. Equivalently, estimate $TV(J_{\mathrm{sim}},J_{\mathrm{real}})$ from samples and check whether it is actually below $\tau$.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that a $\delta$-confident flowpipe $X = \bar{X} \oplus \delta X$ remains valid when the residual is defined in PCA coordinates as $\rho = \max_j(|r_j|/\omega_j)$, where $r = V^\top(PE - \bar{PE})$ projects the prediction errors onto principal axes and $\omega_j$ are the maximum projected magnitudes seen in training. With a separate calibration dataset and robust conformal inference, the order statistic $\rho^*_{\delta,\tau}$ at rank $\ell^* = \lceil (L+1)(1+1/L)(\delta+\tau) \rceil$ bounds the $\delta$-quantile of the real residual distribution as long as the total variation between simulated and real residuals is below $\tau$. Proposition 5 converts that scalar bound into a conjunction of interval constraints on the projected errors, defining an inflating parallelotope $\delta X = \langle \bar{PE}, V, P \rangle$; Lemma 3 then gives the flowpipe $X = \bar{X} \oplus \delta X$. The paper reports that this yields a valid 99.99% flowpipe in the $\tau=0$ cases and a valid 95% flowpipe for the powertrain under a 20% noise-covariance increase, with visibly smaller volume than the axis-aligned hypercube baseline.

Load-bearing premise

The load-bearing premise is that the total variation between the simulator's residual distribution and the real system's residual distribution is no larger than a pre-chosen number $\tau$; in the shifted experiment $\tau = 0.04$ is stated without being estimated or verified, and if the true total variation exceeds $\tau$ the robust conformal bound in Eq. (4) no longer applies.

Editorial extensions

If this is right

For a fixed $\delta$ and shift bound $\tau$, the PCA-oriented inflating parallelotope is contained in or smaller than the axis-aligned origin-centered hypercube of the baseline method, so the resulting safety certificates cover less volume for the same confidence.
The per-segment training strategy removes the need to iterate a single large model over the time horizon, eliminates the associated cumulative error, and enables exact-star reachability on the small ReLU networks, scaling the approach to horizons of 5000 steps.
The distribution-shift guarantee transfers from the simulator to deployment whenever $TV(J_{\mathrm{sim}}, J_{\mathrm{real}}) \leq \tau$, because robust conformal inference inflates the rank by $\tau$ before selecting the quantile; the paper demonstrates this on the 27-dimensional powertrain with $\tau = 0.04$ and $\delta = 0.95$.
The method covers high-dimensional hybrid systems: the powertrain's 4000-step trajectories are handled with 4000 independent segment models and approx-star reachability, producing a flowpipe that still meets the 95% coverage guarantee.

Reading between the lines

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

A practical extension the paper leaves implicit is estimating $\tau$ from data: since the guarantee needs $TV(J_{\mathrm{sim}},J_{\mathrm{real}}) \leq \tau$, one could collect a small set of real trajectories and use a histogram or kernel estimate of total variation to pick $\tau$, rather than asserting it as in the shifted experiment.
The max-norm residual could be replaced by an $\ell^p$ or sorted-quantile norm, which would trade a slightly larger scalar bound for a tighter shape; with strongly correlated errors this might shrink the flowpipe further while keeping the conformal argument intact.
Because the segment models are independent, the algorithm is embarrassingly parallel; choosing segment lengths adaptively from local error variability, rather than fixing $T_q = 1$, could reduce the number of models without losing exact-star efficiency.
A nonlinear coordinate transform, such as a normalizing flow, could replace the linear PCA projection to capture curved error distributions; the cost is that the reachable set would no longer be a parallelotope, so the star-set arithmetic would need a different representation.

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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

I've read PCA-DDReach. The short version: this is a genuine improvement over the Hashemi et al. 2024b method, with two real contributions—PCA-aligned inflating parallelotopes and segment-wise training of small surrogates. The conformal machinery is handled correctly, including a proper split-conformal split. Worth engaging with. One soft spot matters: the distribution-shift experiment relies on an unverified TV bound.

What's new: the residual is defined on PCA-projected errors, so the inflating set is an oriented parallelotope rather than an axis-aligned hypercube. That reduces conservatism in the plotted examples. The segment-wise training is a practical fix: instead of one big network that maps initial state to the whole trajectory, they train N small networks, one per segment. This removes cumulative error, allows exact-star reachability on small models, and parallelizes naturally. Both ideas fit the problem well.

The theory: Proposition 5 is correct. The equivalence between the max residual and the componentwise bound holds, and calibrating on a separate dataset avoids the circularity that would arise from reusing the training set. The paper says that explicitly, so no issue there.

Where it's soft: Experiment 3 is the only distribution-shift demonstration, and its guarantee rests on tau=0.04 as a bound on TV between simulated and real residuals. The text simply asserts this follows from a 20% increase in process-noise covariance. That is not obvious for a max-of-normalized-PCA-residual, and no estimation or confidence interval is given. If the true TV is larger, Eq. (4) doesn't give the advertised confidence. This is a genuine empirical-support gap in the paper's central claim about handling shift. It's fixable—estimate tau empirically or provide a sensitivity analysis—but it needs to be addressed before I'd trust the strong claim. Also, Experiment 2 uses linear interpolation between trained models to generate surrogates; that's a heuristic that should be validated by reporting variance across runs. I'd like to see code/data.

Overall: the method is sound and the experiments are ambitious. The paper deserves a serious referee round. I'd recommend conditional acceptance, with the TV bound and reproducibility as the key conditions. If you work on conformal reachability, it's definitely worth citing after those are addressed.

Referee Report

3 major / 6 minor

Summary. The paper proposes PCA-DDReach, a data-driven statistical reachability method for stochastic dynamical systems. It replaces the axis-aligned inflating hypercube of Hashemi et al. (2024b) with a hypercube oriented along principal components of the prediction-error vectors, and it introduces a segmented training strategy in which independent ReLU surrogate models map the initial state to each trajectory segment. The residual is defined as the maximum normalized PCA-projected error, and robust conformal inference is applied on a separate calibration dataset to obtain a rank ℓ* (Eq. (4)) that accounts for distribution shift measured by a total-variation bound τ. The resulting δ-confident flowpipe is X = ¯X ⊕ δX (Section 3.2). Experiments cover a 12-dimensional quadcopter and a 27-dimensional powertrain, including one distribution-shift case with τ=0.04.

Significance. If the claimed guarantees hold, the main contribution is meaningful: a component-wise calibrated region in PCA coordinates can reduce conservatism relative to axis-aligned hypercubes, and segmented training addresses the scalability bottleneck of a single trajectory-to-trajectory surrogate model. A clear strength is that the paper uses a proper split-conformal protocol: PCA/scaling parameters are fitted on the training set, and the conformal quantile is computed on a separate i.i.d. calibration set, so the derivation in Proposition 5 does not suffer from circularity. However, the distribution-shift certificate in Experiment 3 depends on an unverified total-variation bound, and the claimed conservatism improvement over the baseline is not quantified.

major comments (3)

[Appendix A.3, Eq. (4)] The total-variation bound τ=0.04 is asserted without any estimation, derivation, or confidence statement. The text says that a 20% increase in the process-noise covariance makes 0.04 a valid upper bound, but the residual in Eq. (12) is a maximum of normalized PCA-projected errors, so its distribution under a perturbed noise covariance is a nontrivial function of the trained surrogate and the PCA fit. If TV(J_sim, J_real) exceeds 0.04, the robust conformal rank in Eq. (4) no longer provides Pr[ρ<ρ*]>δ, and the δ-confident flowpipe in Experiment 3 is not justified. Please either estimate τ with a statistically valid upper confidence bound, derive a rigorous bound for this specific shift, or clearly position Experiment 3 as an empirical illustration without a formal guarantee.
[Section 4.1, Table 1] The central claim that the PCA-based inflating hypercube is less conservative than the method of Hashemi et al. (2024b) is not supported by any quantitative measure. Figure 3 is a qualitative projection, and Table 1 reports only runtimes and dataset sizes, not the volume, interval width, or any scalar conservatism metric of the reachable sets. Experiments 2 and 3 have no baseline at all. Please add quantitative comparisons of set sizes for at least Experiment 1, and report volume or other size metrics for the flowpipes in all experiments.
[Section 4] No experiment reports the empirical coverage of the computed flowpipe on hold-out trajectories from the deployment distribution. For a statistical certificate, the paper should report the fraction of test trajectories contained in X over repeated trials; this is especially important in Experiment 3, where validity depends on the unverified τ=0.04. Without such a check, the numerical evaluation cannot confirm that the method delivers its stated δ-confidence in practice.

minor comments (6)

[Section 2.2] There are several typographical issues, e.g., 'In other word' should be 'In other words', and 'upper-bound' should be 'upper bound' in Section 2.3.
[Section 3.1, Eq. (8)] The notation for segmented surrogates F_q(s0; θ_q) is introduced without a formal definition of the output dimension; please clarify that F_q maps to R^{T_q n} and reconcile this with the component notation F_j in Eq. (1).
[Table 1] The table formatting is difficult to parse, e.g., entries such as '# avg runtime| T_trn |' mix dataset sizes and runtimes; please separate these into distinct columns with clear units.
[Section 2.3, Eq. (4)] The conditions under which Eq. (4) is valid should be stated explicitly, in particular that δ+τ<1 and that L is large enough for ℓ*≤L.
[Experiment 2, Eq. (18)] The analytical interpolation strategy for regenerating surrogate models is described only briefly; please state whether calibration residuals are computed for each interpolated model and whether those interpolated models are also used when fitting the PCA/scaling parameters.
[Figure 2] The caption refers to red and green hypercubes but the figure appears to be grayscale; please ensure the visual distinction is clear or adjust the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PCA-fitted residual is calibrated on a separate dataset, so the conformal quantile is not equivalent to a fitted value; the unverified τ=0.04 in Experiment 3 is an empirical-support gap, not a circular reduction.

full rationale

The derivation chain is self-contained. The residual in Eq. (12) is defined using PCA axes, centers, and scaling factors ω_j fitted on the training set, but the paper explicitly states that reusing the training set for conformal inference would violate CI rules, and therefore defines a separate i.i.d. calibration dataset (Definition 4). The robust conformal rank in Eq. (4) is applied only to calibration residuals, so the quantile ρ* is not a re-used training value. Proposition 5's equivalence between the residual threshold and the predicate P is definitional, but the probability bound comes from the conformal guarantee on the calibration set, not from the construction. Lemma 3 is cited from the authors' prior work, but it is an elementary Minkowski-sum argument (σ ∈ F(s0) ⊕ PE with F(s0) ∈ ¯X and PE ∈ δX) and is not an unverified self-citation that forces the result. The comparison experiments use external simulators (quadcopter and powertrain) and the NNV reachability toolbox. The main weakness is that Experiment 3 asserts τ=0.04 as a valid TV bound for a 20% covariance increase without estimating or verifying it; however, this is a missing empirical validation of an input assumption, not a reduction of the claimed prediction to a fitted parameter or to a self-citation. Therefore no circular step is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim does not introduce new physical entities. It introduces data-derived parameters for the residual scoring function, which are fitted to training data but calibrated on a separate dataset. The most consequential assumption is the TV bound.

free parameters (4)

ω_j = max over training dataset of |r_j| for each j
Data-dependent scaling factors in the residual definition (Eq. 13), normalizing each PCA coordinate.
PE_mean_q = mean of prediction error vectors over training dataset per segment (Eq. 10)
Center of the PCA-aligned inflating hypercube, estimated from training residuals.
V_q = eigenvectors of the training error covariance matrix per segment (Eq. 10)
Orientation of the inflating hypercube, estimated from training residuals.
τ (TV bound) = 0.04 in Experiment 3; 0 in others
User-supplied upper bound on total variation between simulated and real residual distributions; required by robust conformal inference, but not empirically estimated.

assumptions (5)

standard math Robust conformal inference quantile formula (Eq. 4) is valid
The paper relies on Cauchois et al. (2024) for the quantile bound under TV shift without reproving it.
domain assumption TV(J_sim, J_real) ≤ τ for the given τ
Required for the robust conformal guarantee in Proposition 5; asserted in Experiment 3 without verification.
domain assumption Calibration trajectories are i.i.d. from D_sim
Needed for conformal inference validity; stated in Definition 4.
domain assumption Initial state distribution W has support on the compact set I
Ensures the surrogate flowpipe over I covers all deployment initial states.
domain assumption NNV star reachability gives a sound over-approximation of the ReLU network image
The surrogate flowpipe correctness relies on the external NNV toolbox (Tran et al., 2020).

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Cite this review

Pith. "Pith review of PCA-DDReach: Efficient Statistical Reachability Analysis of Stochastic Dynamical Systems via Principal Component Analysis." pith.science (2026). https://pith.science/paper/CIPJM6WL

@misc{pith2026250514935,
  author       = {Pith},
  title        = {Pith review of: PCA-DDReach: Efficient Statistical Reachability Analysis of Stochastic Dynamical Systems via Principal Component Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CIPJM6WL}},
  note         = {Machine review of arXiv:2505.14935}
}

read the original abstract

This study presents a scalable data-driven algorithm designed to efficiently address the challenging problem of reachability analysis. Analysis of cyber-physical systems (CPS) relies typically on parametric physical models of dynamical systems. However, identifying parametric physical models for complex CPS is challenging due to their complexity, uncertainty, and variability, often rendering them as black-box oracles. As an alternative, one can treat these complex systems as black-box models and use trajectory data sampled from the system (e.g., from high-fidelity simulators or the real system) along with machine learning techniques to learn models that approximate the underlying dynamics. However, these machine learning models can be inaccurate, highlighting the need for statistical tools to quantify errors. Recent advancements in the field include the incorporation of statistical uncertainty quantification tools such as conformal inference (CI) that can provide probabilistic reachable sets with provable guarantees. Recent work has even highlighted the ability of these tools to address the case where the distribution of trajectories sampled during training time are different from the distribution of trajectories encountered during deployment time. However, accounting for such distribution shifts typically results in more conservative guarantees. This is undesirable in practice and motivates us to present techniques that can reduce conservatism. Here, we propose a new approach that reduces conservatism and improves scalability by combining conformal inference with Principal Component Analysis (PCA). We show the effectiveness of our technique on various case studies, including a 12-dimensional quadcopter and a 27-dimensional hybrid system known as the powertrain.

Figures

Figures reproduced from arXiv: 2505.14935 by the authors.

**Figure 2.** The figure shows the projection of prediction errors for two-dimensional states [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Shows the comparison with Hashemi et al. (2024b). The blue and red borders are [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Shows the projection of our δ-confident flowpipe on each component of the trajectory state. The shaded area are the simulation of trajectories from T trn [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 6.** Figure 6: Shows the comparison of angular velocity of the last rotating mass in presence and absence of the process noise. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Avoiding geometric intersection operations in reachability analysis of hybrid systems

Matthias Althoff and Bruce H Krogh. Avoiding geometric intersection operations in reachability analysis of hybrid systems. In Proceedings of the 15th ACM international conference on Hybrid Systems: Computation and Control, pages 45--54, 2012

work page 2012

[2] [2]

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Stanley Bak and Parasara Sridhar Duggirala. Simulation-equivalent reachability of large linear systems with inputs. In International Conference on Computer Aided Verification, pages 401--420. Springer, 2017

work page 2017

[3] [3]

Neural predictive monitoring

Luca Bortolussi, Francesca Cairoli, Nicola Paoletti, Scott A Smolka, and Scott D Stoller. Neural predictive monitoring. In Runtime Verification: 19th International Conference, RV 2019, Porto, Portugal, October 8--11, 2019, Proceedings 19, pages 129--147. Springer, 2019

work page 2019

[4] [4]

Robust validation: Confident predictions even when distributions shift

Maxime Cauchois, Suyash Gupta, Alnur Ali, and John C Duchi. Robust validation: Confident predictions even when distributions shift. Journal of the American Statistical Association, pages 1--66, 2024

work page 2024

[5] [5]

Conformal prediction regions for time series using linear complementarity programming

Matthew Cleaveland, Insup Lee, George J Pappas, and Lars Lindemann. Conformal prediction regions for time series using linear complementarity programming. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 20984--20992, 2024

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Data-driven reachable set computation using adaptive gaussian process classification and monte carlo methods

Alex Devonport and Murat Arcak. Data-driven reachable set computation using adaptive gaussian process classification and monte carlo methods. In Proc. of ACC, pages 2629--2634, 2020 a

work page 2020

[7] [7]

Estimating reachable sets with scenario optimization

Alex Devonport and Murat Arcak. Estimating reachable sets with scenario optimization. In Learning for dynamics and control, pages 75--84. PMLR, 2020 b

work page 2020

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Data-driven reachability analysis with christoffel functions

Alex Devonport, Forest Yang, Laurent El Ghaoui, and Murat Arcak. Data-driven reachability analysis with christoffel functions. In Proc. of CDC, pages 5067--5072, 2021

work page 2021