{"id":"60d9fa0a-f880-4fa1-a308-ca92a3b370b5","arxiv_id":"2505.12529","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Introduces an H-infinity norm for linear systems with quadratic output and an optimization-based reduced-order modeling algorithm that minimizes it, with a structure-preserving variant for port-Hamiltonian systems.","lead":"The paper defines an H-infinity norm for systems with quadratic output and proposes an optimization scheme that builds smaller surrogate models close in that norm. It gives engineers a new way to reduce the size of simulation models that also track energy-like quadratic quantities.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The additive H-infinity norm is heuristic; Figure 4 shows the proposed method loses to TSIA in time-domain error, and Corollary 2.3 only bounds, not tightens, the actual L2 error.","rationale":"The reader identified the ad hoc additive norm as the weakest assumption and the Figure 4 time-domain results as evidence that the claimed superiority is overstated. My stress test agrees: the norm is explicitly admitted to lack rigorous rationale (Section 3.1 footnote), and Corollary 2.3 gives only an upper bound, not a tight gain characterization. The numerical evidence in Figure 4 directly contradicts the strongest wording of the claim ('clearly outperform') for relevant inputs. This is not an internal inconsistency; the math derivations appear sound. It is a mismatch between the claim and the evidence, and between the objective and the performance measure, that should be fixed by softening the claim and adding a broader benchmark. The verdict CONDITIONAL is appropriate, conditioned on addressing the norm's rationale and the numerical comparison claims.","tokens_in":22068,"tokens_out":1385,"duration_ms":13088,"concrete_test":"Run the proposed algorithm and TSIA on the mass-spring-damper benchmark with a family of inputs spanning different frequency ranges, including a band-limited input concentrated at the frequencies where the proposed method currently wins and a wide-band input where TSIA wins, and compute the L2 output error for each ROM. If the proposed method fails to beat TSIA on a substantial fraction of these inputs, or if the ranking of methods changes with the input class, then the claim that the H-infinity method 'clearly outperforms' state-of-the-art methods is not supported for the actual L2 error objective.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that Algorithm 2 minimizes a meaningful H-infinity norm and thereby outperforms state-of-the-art methods. The load-bearing premise is Definition 2.1, where the H-infinity norm is the sum of separate linear and quadratic transfer-function suprema. The paper itself admits in a Section 3.1 footnote that no rigorous rationale justifies a particular partitioning of linear and quadratic contributions. Theorem 2.2.3 only provides an upper bound on the L2-to-L2 gain; it does not show that minimizing the additive norm minimizes the actual L2 output error for finite inputs. The heuristic nature is not just a theoretical caveat: Figure 4 shows that for inputs with higher-frequency or chirp content, the H2-focused TSIA method yields smaller time-domain output errors than the proposed H-infinity method. Because the claimed superiority rests on the H-infinity error metric on a single example, while the stated goal is small L2 output error for square-integrable inputs, the algorithm's objective is not tightly coupled to the actual performance measure. This is a specific, addressable mismatch rather than a fatal flaw.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a serious referee. The paper defines an H-infinity norm for LTIQO systems as the sum of the linear and quadratic transfer-function suprema, proves it is a norm, obtains an upper bound on the L2-to-L2 gain, and then constructs an optimization-based MOR algorithm with explicit gradients. It also introduces a condensed port-Hamiltonian parametrization that reduces degrees of freedom relative to the prior SOBMOR work. The derivations are careful, the algorithm is nontrivial, and the code is public and reproducible. That is real work.\n\nThe soft spot is the numerical claim. The algorithm minimizes exactly the norm the paper defines, so seeing it win on H-infinity error against H2-based methods is expected, almost by construction. The paper itself says so. The more honest test is Figure 4, where TSIA gets smaller time-domain L2 error for higher-frequency and chirp inputs. That mismatch is not fatal, but the abstract's phrase \"outperforms state-of-the-art\" is too strong.\n\nA deeper issue is the norm itself. The additive combination of linear and quadratic suprema is a heuristic; the footnote in Section 3.1 openly says no rigorous rationale exists for how to weight the two contributions. Theorem 2.2 gives an upper bound, not a tight characterization of the gain, so minimizing this norm may not minimize the output error for finite inputs. This is acknowledged in the text, but it is load-bearing for the approach. The paper's own summary points to investigating alternative weightings, which is the right next step.\n\nThese are addressable, not disqualifying. The definition is new, the analysis is sound as far as it goes, and the structure-preserving parametrization is a genuine improvement. The paper is for people working on model reduction with nonlinear outputs, port-Hamiltonian systems, and H-infinity notions beyond LTI.\n\nMy recommendation: send it to peer review. Ask for more numerical benchmarks, especially time-domain comparisons across a range of inputs, and a fuller discussion of the weighting ambiguity. The paper should be accepted only after that framing is corrected.","headline":"Genuinely new H-infinity norm for quadratic-output systems and a careful MOR algorithm, but the numerical victory lap is over the norm it optimizes; time-domain results are mixed.","tokens_in":22825,"tokens_out":1906,"would_cite":true,"duration_ms":22113,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37J06","37M99","65P10","93A15","93B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces an H-infinity norm for LTIQO systems as the sum of linear and quadratic transfer-function suprema, proves it is a norm and bounds the L2-to-L2 gain, and gives an optimization-based, nonintrusive model reduction…","keywords":["H-infinity norm","quadratic output systems","model order reduction","port-Hamiltonian systems","leveled least squares","balanced truncation","H2-optimal reduction","transfer functions"],"falsifier":"Construct a stable LTIQO system and two reduced models with the same, or smaller, new $\\mathcal{H}_\\infty$ error but different realized $L_2$ output errors for a concrete input; if the model with smaller $\\mathcal{H}_\\infty$ error has larger actual $L_2$ error, then minimizing this norm does not necessarily minimize output error. Concretely, compute for a fixed input $u$ the ratio $\\|y - \\hat{y}\\|_2 / (\\|u\\|_2 + \\|u\\|_2^2)$ and compare it with $\\|\\Sigma - \\hat{\\Sigma}\\|_{\\mathcal{H}_\\infty}$ on a benchmark where the bound is known to be loose; a systematic high-frequency or chirp input sweep would show whether the $\\mathcal{H}_\\infty$ advantage translates to time-domain performance, since the paper's Figure 4 already reports one case where TSIA has lower $L_2$ error.","tokens_in":21805,"feed_emoji":"📉","tokens_out":6757,"duration_ms":64115,"temperature":0.7,"pith_summary":"This paper establishes a meaningful $\\mathcal{H}_\\infty$ norm for linear time-invariant systems with quadratic output (LTIQO systems), where the output depends linearly and quadratically on the state and input. The norm is defined as the sum of the largest singular values of the linear transfer function and of the quadratic transfer function, and it is proved to be a norm, to reduce to the classical $\\mathcal{H}_\\infty$ norm when the quadratic terms vanish, and to bound the $L_2$-to-$L_2$ output gain. On top of this norm, the paper builds a nonintrusive optimization-based model order reduction algorithm that minimizes an $\\mathcal{H}_\\infty$ error measure between the full-order and reduced-order models, and reports that the resulting reduced models beat balanced truncation and an $\\mathcal{H}_2$-focused two-sided iterative algorithm in $\\mathcal{H}_\\infty$ error. A structure-preserving variant for port-Hamiltonian systems is also given. A sympathetic reader would care because this supplies a worst-case, frequency-domain quality measure for a model class that arises in port-Hamiltonian simulation, optimal control, and stochastic problems, where previously only $\\mathcal{H}_2$-type methods were available.","feed_headline":"Quadratic-output models get an H-infinity norm","feed_subtitle":"New norm plus optimization algorithm reduce worst-case error in nonlinear LTI systems with quadratic outputs.","key_machinery":"The central object is the $\\mathcal{H}_\\infty$ norm of Definition 2.1, written as the sum of two suprema: the largest singular value of the linear transfer function $G_1(s)=C(sI-A)^{-1}B+D$ and the largest singular value of the quadratic transfer function $G_2(s_1,s_2)=\\sum_{j=1}^p e_j\\otimes \\operatorname{vec}(K_j(s_1,s_2))^T$ with $K_j(s_1,s_2)=B^T(s_2I-A)^{-T}M_j(s_1I-A)^{-1}B+P_j$. Theorem 2.2 shows it is a norm, that it reduces to the standard $\\mathcal{H}_\\infty$ norm for LTI systems, and that it bounds the $L_2$-to-$L_2$ gain. The optimization machinery replaces the nonsmooth problem of minimizing $\\|\\Sigma-\\hat{\\Sigma}\\|_{\\mathcal{H}_\\infty}$ by a leveled least squares objective over a parametrized reduced model: for sampled frequency grids, each singular value above a level $\\gamma$ is penalized through a ReLU term, with $\\gamma$ updated by bisection and frequencies adapted during the iteration. Gradients are derived from singular-value derivatives and, for the quadratic part, from derivatives of $\\|K_e(s_1,s_2)\\|_F^2$. A condensed parametrization for port-Hamiltonian systems, based on the factorization $\\hat{A}=(\\hat{J}-\\hat{R})\\hat{Q}$ and Proposition 3.3, reduces the degrees of freedom and preserves the Hamiltonian structure in the reduced model.","core_discovery":"The paper's central claim is that LTIQO systems admit an $\\mathcal{H}_\\infty$ norm that is a genuine norm, generalizes the classical LTI norm, and controls time-domain output error, and that this norm can be minimized by a gradient-based algorithm. In detail, Definition 2.1 sets $\\|\\Sigma\\|_{\\mathcal{H}_\\infty} = \\sup_{s\\in\\mathbb{C}^+} \\sigma_{\\max}(G_1(s)) + \\sup_{s_1,s_2\\in\\mathbb{C}^+} \\sigma_{\\max}(G_2(s_1,s_2))$; Theorem 2.2 proves the norm properties and the bound $\\|y\\|_2 \\leq \\|G_1\\|_{\\mathcal{H}_\\infty}\\|u\\|_2 + \\|G_2\\|_{\\mathcal{H}_\\infty}\\|u\\|_2^2$, and Corollary 2.3 transfers the bound to the error system. The optimization algorithm (Algorithm 2) minimizes a leveled least squares surrogate of $\\|\\Sigma - \\hat{\\Sigma}\\|_{\\mathcal{H}_\\infty}$, updating the level $\\gamma$ by bisection and sampling frequencies adaptively. The authors report numerical results on a mass-spring-damper benchmark in which the method 'clearly outperform[s]' balanced truncation and TSIA in $\\mathcal{H}_\\infty$ error, including a structure-preserving port-Hamiltonian variant.","pith_inferences":["Editorial inference: the equal-weight additive norm is a modeling choice; weighting the linear and quadratic parts differently would likely improve time-domain performance on inputs whose energy sits in one frequency regime, a direction the paper explicitly leaves open.","The paper's own Figure 4 shows TSIA has lower $L_2$ error for high-frequency and chirp inputs; an editor would infer the claimed superiority is specific to the $\\mathcal{H}_\\infty$ measure, and the practical gain depends on input spectrum.","Testable extension: apply the same norm and algorithm to stochastic variance outputs or linear-quadratic cost outputs, two classes the paper names as natural LTIQO applications; no additional theory appears needed beyond the single-quadratic-output derivative formulas.","Editorial inference: because the bound in Theorem 2.2 is not tight, two models with the same $\\mathcal{H}_\\infty$ norm can have different true $L_2$-to-$L_2$ gains; a data-driven gain computation could rank such models where the norm cannot."],"forward_implications":["Theorem 2.2 gives every stable LTIQO system a scalar worst-case measure that controls output error, so model reduction results for this class can be compared quantitatively.","Algorithm 2 produces reduced models that, on the tested benchmark, achieve smaller $\\mathcal{H}_\\infty$ error than balanced truncation and TSIA, so frequency-domain worst-case quality is directly optimized rather than approximated.","The nonintrusive formulation requires only transfer function evaluations, so the method can be applied to data-driven or black-box settings where full state matrices are unavailable.","The condensed port-Hamiltonian parametrization reduces degrees of freedom and, in the numerical example, produces better-conditioned system matrices, making structure-preserving $\\mathcal{H}_\\infty$ reduction computationally cheaper.","Corollary 2.3 implies that a small $\\mathcal{H}_\\infty$ error guarantees a small output error in an $L_2$ sense up to constants, giving an error certificate for the reduced model."],"supporting_citations":[{"why":"Supplies the leveled least squares optimization strategy, the parametrization framework, and the starting point that this paper generalizes to quadratic outputs.","marker":"[5]"},{"why":"Gives balanced truncation for quadratic-output systems, serving as the main baseline and as the source of the kernel representation used for the transfer functions.","marker":"[11]"},{"why":"Provides the TSIA $\\mathcal{H}_2$-optimal reduction method for multiple quadratic outputs, the baseline the paper compares against and outperforms in $\\mathcal{H}_\\infty$ error.","marker":"[15]"},{"why":"Motivates the quadratic Hamiltonian output that turns port-Hamiltonian systems into LTIQO systems and justifies the structure-preserving variant.","marker":"[6]"},{"why":"Supplies the adaptive frequency sampling and the bisection level-update strategy incorporated into Algorithm 2.","marker":"[21]"},{"why":"Provides the $(J-R)Q$ factorization used to enforce stability and port-Hamiltonian structure in the reduced models.","marker":"[26]"},{"why":"Supplies the mass-spring-damper benchmark problem used for all numerical experiments.","marker":"[29]"}],"fun_headline_variants":["H-infinity norm for quadratic outputs","New norm tames nonlinear systems","Quadratic models get worst-case error bound","H-infinity reduction outperforms balanced truncation","Minimize worst-case error in quadratic systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the $\\mathcal{H}_\\infty$ norm should be the plain sum of the linear and quadratic transfer function peaks; the paper notes in a footnote that no rigorous rationale currently justifies any particular split of the two contributions.","fun_headline_variants_meta":{"raw":{"variants":["H-infinity norm for quadratic outputs","New norm tames nonlinear systems","Quadratic models get worst-case error bound","H-infinity reduction outperforms balanced truncation","Minimize worst-case error in quadratic systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000933,"raw_usage":{"total_tokens":3996,"prompt_tokens":951,"completion_tokens":3045,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":2981}},"tokens_in":567,"tokens_out":3045,"duration_ms":21372,"temperature":1.0,"reasoning_tokens":2981,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:31:44.405213+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a stable LTIQO system and two reduced models with the same, or smaller, new $\\mathcal{H}_\\infty$ error but different realized $L_2$ output errors for a concrete input; if the model with smaller $\\mathcal{H}_\\infty$ error has larger actual $L_2$ error, then minimizing this norm does not necessarily minimize output error. Concretely, compute for a fixed input $u$ the ratio $\\|y - \\hat{y}\\|_2 / (\\|u\\|_2 + \\|u\\|_2^2)$ and compare it with $\\|\\Sigma - \\hat{\\Sigma}\\|_{\\mathcal{H}_\\infty}$ on a benchmark where the bound is known to be loose; a systematic high-frequency or chirp input sweep would show whether the $\\mathcal{H}_\\infty$ advantage translates to time-domain performance, since the paper's Figure 4 already reports one case where TSIA has lower $L_2$ error.","supporting_citations":[{"cited_title":"$H_2$ optimal model reduction of linear systems with multiple quadratic outputs","cited_arxiv_id":"2405.05951","evidence_quote":"Provides the TSIA $\\mathcal{H}_2$-optimal reduction method for multiple quadratic outputs, the baseline the paper compares against and outperforms in $\\mathcal{H}_\\infty$ error."},{"cited_title":"Energy matching in reduced passive and port-Hamiltonian systems","cited_arxiv_id":"2309.05778","evidence_quote":"Motivates the quadratic Hamiltonian output that turns port-Hamiltonian systems into LTIQO systems and justifies the structure-preserving variant."},{"cited_title":"IFAC-PapersOnLine54(19), 143– 148 (2021) https://doi.org/10.1016/j.ifacol.2021.11.069","cited_arxiv_id":null,"evidence_quote":"Supplies the adaptive frequency sampling and the bisection level-update strategy incorporated into Algorithm 2."},{"cited_title":"Automatica85, 113–121 (2017) https: //doi.org/10.1016/j.automatica.2017.07.047","cited_arxiv_id":null,"evidence_quote":"Provides the $(J-R)Q$ factorization used to enforce stability and port-Hamiltonian structure in the reduced models."}],"review_version":1}