{"id":"fad1051f-7146-45ff-8580-cd6ab7411a6e","arxiv_id":"2411.13404","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper shows δ-covering dissipativity estimation has intractable sample complexity and proposes a PAC generalization bound to quantify robustness of ML-based dissipativity estimation.","lead":"This paper studies methods that estimate dissipativity of nonlinear systems from input-output data. It argues that δ-covering methods need impossibly many samples, while machine learning methods can give practical probabilistic guarantees.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The positive claim in Sec. V depends on a sampling distribution D whose relation to L2e is unspecified; the PAC bound is an average-case guarantee, not the worst-case dissipativity certificate that the Dissipativity Theorem requires.","rationale":"The negative result about δ-covering is a useful tutorial contribution and appears robust despite a flawed norm-containment statement. The positive method, however, only transfers properties from the training distribution to the same D; it does not transfer to L2e, which is what robust dissipativity and the Dissipativity Theorem require. This is not merely an external philosophical worry: the paper's own Fig. 3 shows the Wiener distribution assigning vanishing weight to high-frequency inputs for large T, so the learned dissipativity can miss the lower conic bound on L2e even as K grows. The proposed alternative of mixed time lengths patches one distribution but does not remove the dependence on D. The concern is therefore load-bearing: if unresolved, the abstract's statement that the method 'quantifies the robustness' of ML-based dissipativity estimation overstates what is proven. It is, however, addressable by reframing the contribution as a distribution-dependent average-case guarantee and by clearly separating it from worst-case stability certificates. This matches the reader's weakest assumption, so no change in verdict is needed beyond the conditional acceptance already given.","tokens_in":11170,"tokens_out":14381,"duration_ms":167481,"concrete_test":"Re-run the OC-SVM pipeline (Eq. 8) on Eq. 5 with K = 1000 inputs drawn from the Wiener-process D of Fig. 3 with T=100, compute the PAC bound from [22] (with γ=ρ), and then evaluate the learned sector on a deterministic grid of Fourier inputs whose frequencies lie in the high-frequency range that the Wiener process undersamples. If the PAC bound is small while the grid evaluation violates the learned conic sector, the certificate is confirmed to be distribution-specific; if no grid violation occurs, the concern is weakened for this example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IV's negative result is broadly correct; the stated inclusion B_L2_δ ⊆ B_L∞_δ in the Ω(ρ^b) argument is false (the true inclusion goes the other way up to a √T factor), but a Euclidean volume argument repairs the lower bound, so the intractability conclusion survives. The load-bearing weakness is Section V. Assumption 6 postulates data drawn from a distribution D on Ue, and the Schölkopf et al. bound controls P_{u∼D}(⟨Π,Γ(u)⟩<0) only. Dissipativity (Def. 1) and the Dissipativity Theorem are worst-case: Eq. (1) must hold for all u∈Ue. For any finite K the bound permits a positive-D-measure set of violations and says nothing about inputs outside the support of D. The paper itself concedes 'the meaning of which is not immediately clear.' Neither proposed D resolves this: uniform sampling over b basis coefficients has no well-defined limit over L2e as b→∞, and the Wiener-process D under-samples high-frequency content for large T (Fig. 3), which the paper demonstrates. Thus the claimed tractable trade-off is a statement about a chosen distribution, not a certificate of robustness on the input space relevant to stability.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11392,"tokens_out":7412,"duration_ms":85804,"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":[{"comment":"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.","section":"Section V, Assumption 6"},{"comment":"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.","section":"Section IV.A, Eq. (4)"},{"comment":"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).","section":"Section V, Eq. (8)"}],"minor_comments":[{"comment":"There are several typos: 'cardonality' should be 'cardinality', and Assumption 5 uses 'Lipschiz' instead of 'Lipschitz'.","section":"Section II"},{"comment":"The set-builder notation for U′ is unclear; the sampling grid and the definition of the coefficients α_i should be stated explicitly.","section":"Section IV.A, Eq. (3)"},{"comment":"The Wiener process is consistently misspelled as 'Weiner'; please correct throughout.","section":"Section V and Figures 3-4"},{"comment":"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.","section":"Figure 2 caption"},{"comment":"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.","section":"Appendix, Eqs. (10)-(12)"},{"comment":"Reference [12] lists 'G. Allgöwer' while the other Allgöwer references use 'F. Allgöwer'; this should be corrected.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's tutorial content and negative results are useful, but the positive PAC-based robustness claim is not yet established because the distribution-relative guarantee does not match the worst-case requirement of dissipativity. The inclusion error in Section IV.A should also be corrected. I see no reason to doubt the authors' integrity; the issues are technical and appear addressable within the scope of a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper earns its keep through Section IV. The demonstration that δ-covering methods need K=(ρb)^b samples and can converge to the wrong conic bounds even at δ=0 when the basis set is fixed and uninformative is a real contribution. The Legendre-basis examples and the pendulum case make the point concretely. The appendix's O(1)-space formulas for cone(a,b) are a modest but clean extension of [10]. The citation pattern looks normal, and the authors are honest that the work is primarily tutorial. Credit is due for stating Assumption 3 explicitly and for showing how it interacts with time horizon and basis choice.\n\nThe central sample-complexity argument holds up. There is a technical slip: the claim B_L2_δ ⊆ B_L∞_δ is backwards up to a √T factor, but a Euclidean volume argument repairs the lower bound, so the Ω(ρ^b) conclusion survives. That is a minor issue, not a load-bearing flaw.\n\nThe main weakness is Section V. The OC-SVM plus Schölkopf PAC bound gives a guarantee on P_{u∼D}(<Π,Γ(u)><0). Dissipativity is a worst-case property, and the Dissipativity Theorem needs Eq. (1) to hold for every u in Ue. The paper itself says the meaning of D is not immediately clear, and the two candidate distributions do not fix the problem. Uniform sampling over b basis coefficients has no well-defined limit as b→∞, and the Wiener-process choice under-samples high-frequency content for large T, as Figure 3 shows. So the claimed tractable trade-off is really a distribution-relative statement, not a worst-case certificate. That does not kill the paper, but the abstract and conclusions overstate it. The numerical study uses 1000 samples, which is too few to invoke the stated PAC bound; the authors acknowledge this, but it means Figure 2 is suggestive rather than a demonstration of the bound.\n\nWho is this for? People working in data-driven dissipativity, especially those who need to understand why δ-covering does not scale. It deserves a serious referee: the negative result and the appendix are worth publishing, and the ML section can be revised into a clearer distribution-relative claim. I would send it to review with a request for revision, not desk reject. For a journal, I would want the authors to either prove a worst-case statement or explicitly reframe the ML contribution as distribution-relative, and to provide code or more reproducible experimental details.","headline":"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.","tokens_in":11976,"tokens_out":1689,"would_cite":true,"duration_ms":20147,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For nonlinear data-based dissipativity estimation, dense input grids need super-exponential samples, while PAC generalization bounds make the trade-off tractable.","keywords":["dissipativity estimation","data-based control","nonlinear systems","sample complexity","delta-covering","probably approximately correct learning","one-class support vector machine","conic sectors"],"falsifier":"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.","tokens_in":10889,"feed_emoji":"⚙️","tokens_out":7870,"duration_ms":82693,"temperature":0.7,"pith_summary":"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.","feed_headline":"Super-exponential data cost for dissipativity; PAC bounds fix it","feed_subtitle":"Probabilistic ML certificates keep sample counts flat as input basis size grows, unlike dense grid sampling.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"defines (Q,S,R)-dissipativity and the Dissipativity Theorem that gives the stability motivation.","marker":"[5]"},{"why":"provides the persistent-excitation foundation that enables the LTI simplification, contrasting with the nonlinear case.","marker":"[6]"},{"why":"introduces the δ-covering sampling scheme and Lipschitz-based dissipativity bounds whose sample count is analyzed.","marker":"[10]"},{"why":"extends δ-covering to general dissipativity via extreme ray enumeration, the method whose complexity is compared.","marker":"[11]"},{"why":"Gaussian-process variant of δ-covering; the paper notes it also struggles with the curse of dimensionality for large b.","marker":"[12]"},{"why":"introduces the OC-SVM dissipativity learning formulation that the paper reuses for PAC robustness quantification.","marker":"[13]"},{"why":"presents the ML-based dissipativity learning framework and uniform Fourier-coefficient sampling baseline.","marker":"[15]"},{"why":"supplies the PAC learning definitions and the framework behind Assumption 6.","marker":"[16]"},{"why":"supplies the OC-SVM generalization error bound used to certify robustness without scaling in b.","marker":"[22]"}],"fun_headline_variants":["Grid sampling blows up; ML dissipativity certificates scale","PAC bound fixes dissipativity's super-exponential data cost","Avoid δ-cover explosion: ML dissipativity with flat sample count","Super-exponential grid cost; PAC bounds flatten it"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Grid sampling blows up; ML dissipativity certificates scale","PAC bound fixes dissipativity's super-exponential data cost","Avoid δ-cover explosion: ML dissipativity with flat sample count","Super-exponential grid cost; PAC bounds flatten it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000183,"raw_usage":{"total_tokens":1301,"prompt_tokens":922,"completion_tokens":379,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":312}},"tokens_in":538,"tokens_out":379,"duration_ms":4794,"temperature":1.0,"reasoning_tokens":312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:28:42.783705+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dissipative dynamical systems: b asic input- output and state properties,","cited_arxiv_id":null,"evidence_quote":"defines (Q,S,R)-dissipativity and the Dissipativity Theorem that gives the stability motivation."},{"cited_title":"A note on persistancy of excitation,","cited_arxiv_id":null,"evidence_quote":"provides the persistent-excitation foundation that enables the LTI simplification, contrasting with the nonlinear case."},{"cited_title":"Some problems aris ing in controller design from big data via input-output methods,","cited_arxiv_id":null,"evidence_quote":"introduces the δ-covering sampling scheme and Lipschitz-based dissipativity bounds whose sample count is analyzed."},{"cited_title":"Determi ning dis- sipation inequalities from input-output samples,","cited_arxiv_id":null,"evidence_quote":"extends δ-covering to general dissipativity via extreme ray enumeration, the method whose complexity is compared."},{"cited_title":"Data-driven inf erence of passivity properties via Gaussian process optimization,","cited_arxiv_id":null,"evidence_quote":"Gaussian-process variant of δ-covering; the paper notes it also struggles with the curse of dimensionality for large b."},{"cited_title":"Input-output data-driven co ntrol through dissipativity learning,","cited_arxiv_id":null,"evidence_quote":"introduces the OC-SVM dissipativity learning formulation that the paper reuses for PAC robustness quantification."},{"cited_title":"Dissipativity learning control (DLC): Theoretic al foundations of input-output data-driven model-free control,","cited_arxiv_id":null,"evidence_quote":"presents the ML-based dissipativity learning framework and uniform Fourier-coefficient sampling baseline."},{"cited_title":"Shalev-Shwartz and S","cited_arxiv_id":null,"evidence_quote":"supplies the PAC learning definitions and the framework behind Assumption 6."},{"cited_title":"Estimating the support of a high-dimensional distribu- tion,","cited_arxiv_id":null,"evidence_quote":"supplies the OC-SVM generalization error bound used to certify robustness without scaling in b."}],"review_version":1}