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Issues with Input-Space Representation in Nonlinear Data-Based Dissipativity Estimation

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For nonlinear data-based dissipativity estimation, dense input grids need super-exponential samples, while PAC generalization bounds make the trade-off tractable.

desk verdict A genuinely useful tutorial with a solid negative result on δ-covering; the ML positive claim needs a sharper statement about the sampling distribution before it supports worst-case dissipativity. read the letter →

arxiv 2411.13404 v1 pith:GL5WIDH6 submitted 2024-11-20 eess.SY cs.SYmath.OC

Ethan LoCicero , Alex Penne , Leila Bridgeman This is my paper · ORCID
classification eess.SYcs.SYmath.OC
keywords dissipativityestimationdata-basedcontrolnonlinearsystemssamplecomplexitydelta-coveringprobablyapproximatelycorrectlearningone-classsupportvectormachineconicsectors
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 argues that the standard way of guaranteeing dissipativity from data for nonlinear systems—sampling a dense grid of inputs called a δ-covering—cannot be made both practical and reliable. The sample count grows as $K=(\rho b)^b$, so covering a realistic input space requires super-exponential data, and when the number of basis functions $b$ is kept small to stay feasible, the chosen basis can miss the inputs that determine the true dissipativity bounds. The authors propose instead to treat dissipativity estimation as a machine-learning problem and to certify it with a probably-approximately-correct (PAC) generalization bound on a one-class support vector machine. That bound makes the required sample count independent of the number of basis functions, so the input space can be approximated much more closely without extra data. If the argument holds, dense grid search is the wrong tool for nonlinear data-based dissipativity, and probabilistic ML certificates with explicit distributional assumptions are the practical path.

What carries the argument

The argument runs on two objects. One is the δ-cover of the input space $U_{A1234}$—the set of inputs that are bounded in amplitude, bounded below in norm, have a long enough horizon, and are represented by $b$ orthonormal basis functions—whose sample count $K=(\rho b)^b$ is what makes the robustness guarantee cost super-exponential. The other is the PAC generalization error $|L_D - L_S|$ for the one-class SVM that fits the half-plane $\langle \Pi, \Gamma(u) \rangle \ge 0$, bounded by the cited one-class SVM generalization theory; this bound does not grow with $b$, so it decouples the sample complexity from how finely the input space is approximated. Also load-bearing are the conic-sector formulas in the appendix, which compute cone$(a,b)$ with $O(1)$ space, turning sampled trajectories into concrete dissipativity bounds.

What would settle it

Run the δ-covering estimator on the system $\hat{G}(\omega)=1/(j\omega+1)+1/4$ with the first four Legendre basis functions and $T=10$, sending $K$ to infinity; the paper reports the estimated cone converges to cone(0.68, 1.25) instead of the true cone(1/4, 5/4), directly demonstrating the failure of the robustness claim for practical basis counts. For the ML side, train the OC-SVM on samples from one distribution $D$ and test on inputs drawn from a different distribution; if the error exceeds the PAC bound, then the distributional premise is doing the work and the robustness guarantee is conditional rather than worst-case.

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Extended reading notes

Core claim

The central claim is that δ-covering methods for input-space representation in nonlinear data-based dissipativity estimation face an intractable trade-off: either the sample complexity is too large to implement, or the robustness guarantee does not extend to the true input space $L_{2e}$. For a covering radius $\delta$ and sampling density $\rho$, the number of samples is $K=(\rho b)^b$, so $K$ is $O(b^b)$ for constant density; and because an $L_\infty$ cover requires at least $\rho^b$ points, every δ-covering scheme has super-exponential complexity in $b$. Existing case studies use $b \le 5$, and the paper shows that with small $b$ the estimated dissipativity can converge to wrong conic bounds even for a known linear system: with the first four Legendre bases and $T=10$, the estimate for the system in Equation (5) converges to cone(0.68, 1.25) rather than the true cone(1/4, 5/4). As an alternative, the paper proposes quantifying the robustness of ML-based dissipativity estimation using a PAC generalization bound for the one-class SVM problem; the bound does not scale with $b$ or input/output dimensions, so a large number of basis functions can be used while keeping the sample complexity fixed.

Load-bearing premise

The load-bearing premise is Assumption 6: the input trajectories used in training are drawn from the same probability distribution $D$ that describes the inputs the system will actually see, and the PAC guarantee says nothing about inputs that are not drawn from that distribution.

Editorial extensions

If this is right

  • For any δ-covering scheme, the sample count is at least $\rho^b$ in the $L_\infty$ norm and $O(b^b)$ at constant density, so robust nonlinear dissipativity estimation by grid search is out of reach for input spaces large enough to approximate $L_{2e}$.
  • Keeping $b$ small to make δ-covering feasible changes which inputs are represented, and the estimated dissipativity can converge to wrong values even in the infinite-data limit; data collection and basis choice must be aligned with the true input space.
  • Using the OC-SVM PAC bound, the number of samples needed to certify dissipativity with high probability does not grow with the number of basis functions, so input representations with $b=100$ or more become practical.
  • Uniformly sampling bounded basis coefficients and using Wiener-process inputs with a range of time horizons both yield good conic-sector estimates in the case studies, the latter without requiring amplitude or norm bounds.
  • A single Wiener process trimmed at multiple horizons can estimate dissipativity directly on truncated signals, removing the assumption that a long enough horizon guarantees dissipativity.

Reading between the lines

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

  • The PAC certificate depends on the distribution $D$, so if the inputs that threaten closed-loop stability are rare or adversarial, the probabilistic guarantee can be arbitrary; a natural extension is to shape $D$ to cover the frequency ranges that matter for stability, or to combine the certificate with adversarial search over basis coefficients.
  • The super-exponential sample count in $K=(\rho b)^b$ suggests that any input-space covering approach, not just δ-covering, will fail for high-dimensional bases; the ML route reframes the problem from covering to coverage, which is why it scales.
  • The $O(1)$-space conic formulas in the appendix could be used as streaming estimators, updating dissipativity bounds incrementally as data arrive, which would fit online data-based control loops.
  • A testable extension: for a nonlinear system whose dissipativity is determined by a narrow frequency band, compare the PAC certificate trained on uniform basis-coefficient sampling against one trained on a distribution that emphasizes that band; the certificate should tighten, suggesting $D$ should encode prior physical knowledge.
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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 / 6 minor

Summary. This manuscript is a tutorial-style study of data-based dissipativity estimation for nonlinear systems. It formalizes the assumptions commonly used to restrict the input space L2e to a finite-dimensional set UA1234, then analyzes two families of methods. For δ-covering methods (Section IV), it derives the sample count K=(ρb)^b, argues that this is super-exponential in the number of basis functions b, and gives numerical examples showing that the estimated conic bounds depend strongly on the choice of basis and time horizon. For machine-learning methods (Section V), it proposes applying a PAC generalization bound from Schölkopf et al. to an OC-SVM formulation of dissipativity, claiming that this yields a probabilistic robustness guarantee whose sample complexity is independent of b. Numerical experiments with an LTI system and a pendulum illustrate both the basis-dependence of δ-covering and the behavior of randomly sampled inputs.

Significance. The negative result on δ-covering methods is a useful tutorial contribution: the paper makes explicit that the super-exponential sample complexity, not just the input dimension, is the binding limitation, and the Legendre-basis examples in Figure 1 concretely demonstrate basis-dependent failure even for b=4. The proposed PAC-based approach is a natural and promising direction, and the observation that the generalization bound does not explicitly grow with b is valuable. However, the positive claim is not established as stated: the PAC guarantee is distribution-relative rather than worst-case, and it does not yet provide the certificate required by the Dissipativity Theorem. The manuscript also contains a technical error in the lower-bound argument of Section IV.A, although that conclusion appears repairable. Overall, the paper is likely to be of interest to the data-driven control community, but the central robustness claim needs substantial revision before the manuscript can be accepted.

major comments (3)
  1. [Section V, Assumption 6] The PAC bound displayed in Section V controls P_{u∼D}(⟨Π,Γ(u)⟩<ρ−γ) for a distribution D, but Definition 1 and the Dissipativity Theorem require Eq. (1) to hold for every u∈U_e. The paper concedes that the meaning of D 'is not immediately clear.' Neither proposed interpretation resolves the gap: uniform sampling over b basis coefficients has no well-defined limit in L2e as b→∞, and the Wiener-process sampling demonstrably under-represents high-frequency signals for large T (Figure 3). Therefore, the claimed 'tractable trade-off between robustness and sample complexity' is at present a statement about a chosen sampling distribution, not a certificate of robustness on the input space relevant to stability. The paper should either restrict the claim to a clearly defined distributional robustness notion and show that this suffices for the intended closed-loop stability guarantee, or construct a D whose support equals UA1234 (with a well-defined limit as b,T→∞) and prove the corresponding guarantee.
  2. [Section IV.A, Eq. (4)] The lower-bound argument uses the statement B_L2_δ ⊆ B_L∞_δ. This inclusion is false for continuous-time signals: an L2 ball of radius δ can contain a narrow spike of amplitude M with L2 norm δ and L∞ norm M, where M can be much larger than δ (take support length δ^2/M^2). The correct inclusion is B_L∞_δ ⊆ B_L2_{√T δ} up to a horizon-dependent factor, so the derivation as written is invalid. The Ω(ρ^b) lower bound can likely be recovered by an L2 covering-number or volume argument in the b-dimensional coefficient space, and the paper's negative conclusion may survive, but the proof needs to be corrected because the current inclusion is load-bearing for the claim that all δ-covering methods are intractable.
  3. [Section V, Eq. (8)] The cited generalization bound from [22] is applied to the solution of the hard OC-SVM problem (8), but the manuscript does not verify that the hypotheses of the theorem in [22] hold for this learned solution. In particular, the feature map Γ(u) is not normalized, the kernel associated with the linear product ⟨Π,Γ(u)⟩ is not specified, and the constant c=10^3 is imported without justification. Because this bound is the central evidence for the claimed b-independent sample complexity, the authors should either verify that the solution of (8) satisfies the conditions of the theorem, or derive a suitable bound directly for the hypothesis class defined by (8).
minor comments (6)
  1. [Section II] There are several typos: 'cardonality' should be 'cardinality', and Assumption 5 uses 'Lipschiz' instead of 'Lipschitz'.
  2. [Section IV.A, Eq. (3)] The set-builder notation for U′ is unclear; the sampling grid and the definition of the coefficients α_i should be stated explicitly.
  3. [Section V and Figures 3-4] The Wiener process is consistently misspelled as 'Weiner'; please correct throughout.
  4. [Figure 2 caption] The caption does not explain how uniform sampling over UA1234 is implemented for different b, nor what exactly is plotted as the 'upper bound' and 'lower bound' in each panel.
  5. [Appendix, Eqs. (10)-(12)] The ± notation inside M(u_i) and in the subsequent formulas is ambiguous; the text should clarify that both sign choices are evaluated and the extremal value is selected.
  6. [References] Reference [12] lists 'G. Allgöwer' while the other Allgöwer references use 'F. Allgöwer'; this should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the PAC certificate is an external, distribution-relative bound; the sole same-author citation is a non-load-bearing appendix reference.

full rationale

The paper's central negative claim (Sec. IV) is a counting/volume bound on the size of delta-covers, derived from the definition of a delta-cover and the sampling scheme in Eqs. (3)-(4); it is not an output fitted into the claim. The positive claim (Sec. V) applies an external PAC generalization bound from Schoelkopf et al. [22] to the OC-SVM formulation from [13]. The bound is a standard conditional guarantee: if training data are drawn from D, then with high probability future D-draws are correctly classified. The paper explicitly concedes the limitation: 'This generalization error is only achieved if the data is sampled according to distribution D, the meaning of which is not immediately clear.' That is a distributional premise and a mismatch with the worst-case Dissipativity Theorem, but not a circular reduction: the guarantee is not defined in terms of the conclusion it is used to support. The only same-author citation is [23] (Bridgeman and Forbes) in the appendix, used for two known ways to compute extremal conic bounds; it is not load-bearing for the intractability or PAC results. Technical concerns such as the L2/L-infinity ball inclusion direction in Sec. IV and the average-case versus worst-case gap in Sec. V are correctness risks, not circularity. No step reduces to its inputs by construction.

Assumptions & free parameters 4 free parameters · 9 assumptions · 0 invented entities

The central claims depend on user-chosen parameters (δ, L, γ, b, T, ū, ε) and on Assumptions 1-6. No new physical entities are introduced. The most fragile item is the distribution D in Assumption 6, and the lower-bound proof also uses a questionable norm-containment.

free parameters (4)
  • Covering radius δ
    δ-covering methods require choosing δ; the robustness guarantee and sample count K=(ρb)^b both depend on it. Exactness requires δ→0, making K intractable.
  • Lipschitz constant L
    Assumption 5 requires a known bound on L; the conic-bound computations in Appendix A use L to inflate sample bounds. An incorrect L invalidates the deterministic robust guarantee.
  • PAC margin γ = γ=ρ
    Section V chooses γ=ρ to make the generalization event equal to ⟨Π,Γ(u)⟩<0; the bound's RHS depends on γhat=ρ/||Π||_F, so this choice affects the sample-complexity guarantee.
  • Input-space parameters b, T, ū, ε
    Assumptions 1-4 define U_A1234; the paper shows that choices of b and T determine whether estimated dissipativity matches L2e behavior (e.g., Legendre bases with T=10 vs T=1). These are chosen, not fitted.
assumptions (9)
  • domain assumption Assumption 1: the amplitude of permissible inputs is absolutely bounded above by ū.
    Section III; used to define U_A1234 and the sampling region. Physical but restricts inputs away from L2e.
  • domain assumption Assumption 2: the induced norm of permissible inputs is bounded below by ε.
    Section III; needed because arbitrarily small inputs cannot be densely sampled, but it excludes inputs near zero from the robust guarantee.
  • ad hoc to paper Assumption 3: if Eq. 1 holds for one sufficiently large T, it holds for all T.
    Section III; the authors state this has not been explicitly stated in the literature. It approximates ultimate virtual dissipativity and is needed to extend finite-horizon checks to the infinite horizon; it can fail for finite T.
  • domain assumption Assumption 4: the input signal can be represented by a finite number b of orthonormal basis functions.
    Section III; justifies finite-dimensional sampling. As b→∞ the space approaches L2e, but all practical guarantees are for finite b.
  • domain assumption Assumption 5: the unknown operator G is Lipschitz continuous with known constant L.
    Section IV; required by δ-covering methods and by the conic-bound formulas in Appendix A. A loose or wrong L breaks the deterministic robust bound.
  • domain assumption Assumption 6: data u_i are sampled i.i.d. from a probability distribution D over Ue.
    Section V; the PAC generalization bound from [22] requires it. The paper acknowledges that the meaning of D is not immediately clear.
  • ad hoc to paper The inclusion B_L2_δ ⊆ B_L∞_δ used to lower-bound sample complexity.
    Section IV-A; this is false for signal norms in general, since sup-norm can exceed L2 norm. It is at best valid in coefficient space up to a basis-dependent constant, so the Ω(ρ^b) bound is not rigorously established as written.
  • standard math Parseval's theorem and Fourier series representation of orthonormal basis functions.
    Section IV-B; used in the demonstration that extremal gains are only realized when basis frequencies match Nyquist extremes.
  • standard math The one-class SVM generalization bound of [22].
    Section V; the paper imports this bound with c=10^3, without verifying its hypotheses for the dissipativity feature map.

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Cite this review

Pith. "Pith review of Issues with Input-Space Representation in Nonlinear Data-Based Dissipativity Estimation." pith.science (2026). https://pith.science/paper/GL5WIDH6

@misc{pith2026241113404,
  author       = {Pith},
  title        = {Pith review of: Issues with Input-Space Representation in Nonlinear Data-Based Dissipativity Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GL5WIDH6}},
  note         = {Machine review of arXiv:2411.13404}
}
read the original abstract

In data-based control, dissipativity can be a powerful tool for attaining stability guarantees for nonlinear systems if that dissipativity can be inferred from data. This work provides a tutorial on several existing methods for data-based dissipativity estimation of nonlinear systems. The interplay between the underlying assumptions of these methods and their sample complexity is investigated. It is shown that methods based on delta-covering result in an intractable trade-off between sample complexity and robustness. A new method is proposed to quantify the robustness of machine learning-based dissipativity estimation. It is shown that this method achieves a more tractable trade-off between robustness and sample complexity. Several numerical case studies demonstrate the results.

Figures

Figures reproduced from arXiv: 2411.13404 by the authors.

Figure 1
Figure 1. Left: K = 625 trajectories of Equations 5 (LTI) and 6 (nonlinear) generated with a b = 4 non-sequential Fourier bases  1 T , √ 2 T sin( 2π T t), √ 2 T sin( 20π T t), √ 2 T sin( 200π T t) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Bounds on 1000 trajectories of Equation 5 uniformly s [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Data generated from a single Weiner process applied t [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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