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Adaptive Reduced Order Modelling of Discrete-Time Systems with Input-Output Dead Time

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Dead time splitting by linear programming makes data-driven acoustic model reduction scalable and accurate.

desk verdict Genuinely new DTS formulation and the largest ERA benchmark to date, but the LP's optimality claim fails when m≠p and the delay-matrix estimation is undocumented—worth a serious referee nonetheless. read the letter →

arxiv 2506.08870 v1 pith:LMDNOAVB submitted 2025-06-10 math.DS

Art J. R. Pelling , Ennes Sarradj This is my paper
classification math.DS MSC 93B1593B3093A1593C5568W20
keywords modelorderreductionEigensystemRealizationAlgorithmdeadtimesplittingrandomizedSVDleave-one-outerrorestimatorroomimpulseresponseMIMOstate-spacesystemsinput-output
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 tackles a practical obstacle in building efficient surrogate models from measured room impulse responses: propagation delays, called dead times, inflate the order of state-space models and cause pre-ringing artifacts. The authors propose a dead time splitting (DTS) scheme that solves a linear program to separate each transmission path's delay into common input and output dead times, so that only residual delays remain in the data before model reduction. They also turn the Eigensystem Realization Algorithm into an adaptive randomized pipeline, using a randomized leave-one-out error estimator to choose the model order on the fly and a memory-efficient QR update to reuse computations. On large measured datasets, the DTS-based pipeline yields reduced order models with smaller error at fewer degrees of freedom than previous approaches. If correct, this makes data-driven model reduction practical for very large acoustic measurement sets.

What carries the argument

The load-bearing object is the DTS linear program, Eq. (19). It takes the per-path delay matrix $\delta$ and finds nonnegative input and output dead times $\tau, \theta$ maximizing their total under the constraint that $\theta_i + \tau_j$ does not exceed $\delta_{ij}$, ensuring that no dynamics are pruned when the data are shifted by Eq. (16). The second piece is the adaptive randomized ERA pipeline: randomized SVD with power iterations, a leave-one-out estimator that predicts the RSVD error without forming reduced models, and a communication-avoiding shifted CholeskyQR with a column-update formula so that extra random samples refine the basis without recomputation. The third piece is the correction of the classical $H_2$ error bound, replacing the squared norm with the norm itself and thereby fixing the asymptotic decay; the dead time operators are realized as unitary delay lines whose Hankel singular values are all one, which explains why unshifted dead times are so costly for ERA.

What would settle it

Run the full pipeline on synthetic MIMO systems with known input and output dead times and added measurement noise; if the LP does not recover the true $\tau$ and $\theta$ to within the expected resolution, or if pre-ringing persists in the structured ROM when the delay estimates are deliberately biased, the central claim is refuted.

Watch

Extended reading notes

Core claim

The central claim is that dead time extraction should be formulated as a rank-one delay-structure splitting problem rather than per-channel delay removal. Given per-path dead times $\delta_{ij}$, the paper defines the DTS linear program $\arg\max_{x\ge 0} \|x\|_1$ subject to $Fx \le \mathrm{vec}(\delta)$, where $F$ encodes the constraints $\theta_i + \tau_j \le \delta_{ij}$. Solving it yields input dead times $\tau$ and output dead times $\theta$; shifting the impulse response data by $\tau_j + \theta_i$ and reapplying the dead time operators after ERA produces a structured reduced order model whose dynamical part has lower order and fewer degrees of freedom. The paper further claims that equipping randomized ERA with a randomized leave-one-out error estimator and a CholeskyQR-based updating scheme makes the model order adaptive and scalable to Hankel matrices with millions of columns, and it identifies and corrects a missing square in the classical $H_2$ error bound for ERA.

Load-bearing premise

The per-path dead time matrix $\delta$ is treated as known before solving the linear program, but the paper does not specify how these dead times are estimated for the benchmark datasets; if the estimates are biased, the constraint $Fx \le \mathrm{vec}(\delta)$ is wrong and the claimed degree-of-freedom savings collapse.

Editorial extensions

If this is right

  • With DTS dead time extraction, the same relative model error is reached with fewer degrees of freedom; for MIRD scenarios the saving is about twice that of least-common-dead-time removal.
  • Structured ROMs built from rectified data avoid the pre-ringing artifacts that appear when ERA is applied to unshifted data.
  • The adaptive pipeline makes model order selection automatic: the LOO estimator tracks the true error decay and can be used to set the RSVD tolerance without constructing candidate models.
  • The corrected $H_2$ bound is no longer violated by the benchmark ROMs, while the previously used uncorrected bound overestimates error increasingly with model order.
  • The benchmark sizes, including a Hankel matrix of dimension $65536 \times 1115136$, represent the largest ERA application to date in terms of input data dimension and constructed model order.

Reading between the lines

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

  • Beyond the paper: if the true per-path delays are not exactly rank-one, the LP leaves residual delays that still limit the achievable order; quantifying the residual $\|Fx - \mathrm{vec}(\delta)\|_1$ could serve as a diagnostic for model structure.
  • Beyond the paper: the same input-output delay splitting idea could be applied to other data-driven realization methods beyond ERA, such as balanced truncation of FIR filters, whenever per-channel delay estimates are available.
  • Beyond the paper: the DTS pipeline's claimed benefit depends on the quality of the initial per-path delay estimates, so a reproducible comparison of time-delay-estimation methods feeding the LP would be a direct test of the approach.
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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

2 major / 5 minor

Summary. The paper develops a data-driven reduced-order modelling pipeline for discrete-time MIMO LTI systems with input-output dead times, based on measured impulse responses. The main contributions are: (i) a dead-time splitting (DTS) linear program that separates per-path delays into input and output dead times; (ii) an adaptive randomized ERA that uses a leave-one-out error estimator and a communication-avoiding QR update; (iii) a correction to the classical ERA error bound attributed to a typo in Kung's original paper; and (iv) a large-scale benchmark on the MIRACLE and MIRD room impulse response databases, with open-source code. The paper reports that DTS-based extraction outperforms least-common-dead-time extraction and no extraction in terms of model degrees of freedom versus relative H2 error.

Significance. If the claims hold, this would be the largest practical demonstration of ERA in terms of input data dimension and model order, and it offers a scalable, reproducible pipeline for constructing structured state-space surrogates from measured acoustic impulse responses. The open-source implementation and the careful correction of the ERA error bound are concrete strengths. However, the central methodological claim about the DTS objective is algebraically incorrect as stated, and the estimation of the delay matrix that feeds the LP is never specified. Both issues are load-bearing for the reported empirical gains, so the significance is currently contingent on a corrected formulation and a reproducible delay-estimation step.

major comments (2)
  1. [Section 3.3, Eq. (19)] The statement following Eq. (19), 'The sum of the residual dead times ∥Fx − vec(δ)∥1 is minimized by Eq. (19),' is incorrect when the number of outputs p differs from the number of inputs m. For x = [θ; τ] ≥ 0 and Fx ≤ vec(δ), the residual sum equals Σ_{i,j} δ_ij − p Σ_i θ_i − m Σ_j τ_j, whereas the LP maximizes ∥x∥1 = Σ_i θ_i + Σ_j τ_j. These objectives coincide only when p = m. Every reported benchmark has p ≠ m (MIRACLE: 1024 inputs vs 64 outputs; MIRD: 26 vs 8), so the claimed optimality fails in the regime of all experiments. Moreover, the feasible polyhedron can have multiple optima with the same ∥x∥1 but different residual sums; a minimal example is p=1, m=2, δ=[5,3], where θ=2, τ=[3,1] gives residual zero and θ=0, τ=[3,3] has the same objective value with residual 2. No tie-breaking rule is given. This undermines the stated mechanism behind the improvements in Fig. 8. The fix is straightforward: replace the objective with the weighted sum pΣθ + mΣτ (equivalently, minimize the residual), or add a tie-breaking rule, and re-run the benchmarks; alternatively, the paper should explicitly present ∥x∥1 maximization as a heuristic and justify it empirically rather than by the residual-minimization claim.
  2. [Section 3.4, step 1] The procedure begins with 'Given IR measurement data and the (estimated) dead times between each source and receiver,' but the manuscript nowhere specifies how the delay matrix δ is estimated for the MIRACLE or MIRD datasets. This is not a minor omission: the constraint Fx ≤ vec(δ) and the rectification step Eq. (16) both depend on δ, and a biased δ can make Eq. (16) non-causal or leave excess residual delay, directly affecting the degree-of-freedom savings claimed in Section 5.2. The authors should state the time-delay estimation method used (e.g., onset detection, thresholding, cross-correlation), or provide code or a reference that makes this step fully reproducible.
minor comments (5)
  1. [Abstract and affiliation] The abstract contains the typo 'propsed' and the affiliation line has 'Universti¨at'; both should be corrected.
  2. [Section 3.3, Eq. (21)] The text states 'Since 1 < η_k ≤ s for all k', but η_0 = 1 by definition, so the inequality should read '1 ≤ η_k ≤ s'.
  3. [Section 2.5] In the sentence defining the dead time degrees, the second occurrence of '¯τ' should be '¯θ': it currently reads '¯τ := Pp i=1 θi'.
  4. [Algorithm 2] The input list of Algorithm 2 uses the symbol q for the block size while the body uses b and also uses q for the number of power iterations; this overloading of q is confusing and should be resolved.
  5. [Section 5.2] The comparison in Fig. 8 evaluates ROMs on the same impulse responses used to build them; this is standard in system identification, but the in-sample nature of the 'more accurate' claim should be stated explicitly in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the ROM construction, dead-time splitting, and error estimation are benchmarked on public/external datasets and do not reduce to their own inputs by construction.

full rationale

The paper's central contribution is an algorithmic pipeline: given per-path delays δ (taken as an input in Section 3.4, step 1), solve the DTS linear program Eq. (19) to split delays into input and output dead times, remove them by Eq. (16), run randomized ERA, and reattach the delay structure. The accuracy metric Eq. (25) compares the ROM against the same impulse-response data used for identification, but this is standard in-sample model-reduction assessment, not a "prediction" in the circularity sense; no fitted parameter is renamed as a derived result. The MIRACLE dataset originates from the authors' prior work, but it is a publicly released measured dataset and MIRD is external; citing these datasets is not a load-bearing self-citation that forces the conclusions. The corrected Kung bound is an independent mathematical claim about [43] and does not define the ROM output. The paper itself flags the error estimator as heuristic and notes that the dead-time matrix is an input to the procedure; both are limitations, not instances of conclusion-equals-premise. The skeptic's objection to Eq. (19) — that maximizing ‖x‖₁ does not generally minimize ‖Fx − vec(δ)‖₁ when p ≠ m — is a possible correctness or optimality flaw, not circularity, because the LP objective is not definitionally identical to the residual dead time. No step was found where X is defined in terms of Y and then Y is "derived" from X.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central novel object is the DTS linear program, which introduces no new physical entity. The method rests on assumptions that the measured IRs are LTI Markov parameters with additive input/output dead times and known per-path delays. Numerical hyperparameters (q, b, gamma) are hand-set or unreported. The text line 'delta = theta tau^T' in Section 3.3 is a typo; the equations use theta_i + tau_j.

free parameters (3)
  • Number of power iterations q = 2
    Fixed to q=2 in all experiments based on prior numerical experience [16,32]; affects RSVD accuracy and is chosen by hand, not by an automatic rule.
  • RSVD block size b = not reported
    Algorithm 1 and 2 take b as input, but no value or selection rule is reported for the benchmarks; this affects the adaptive sampling loop.
  • Approximation tolerance gamma = not reported
    Algorithm 1 takes gamma > 0 as input, but the value used for the adaptive stopping criterion is not stated.
assumptions (5)
  • domain assumption Measured room impulse responses are outputs of a causal, stable, discrete-time LTI system whose IR provides the Markov parameters.
    Section 2.2 and Eq (4): the whole method treats measured IRs h_k as Markov parameters of an LTI system.
  • domain assumption The Markov parameters decay to zero beyond the first 2s-1 samples, so the Hankel truncation in Eq (5) is valid.
    Imposed by Eq (9) in Section 2.4; the RIRs are truncated to T60, so the tail is neglected.
  • domain assumption After dead time removal, the residual system is a low-order LTI system G0, and the delay structure is exactly additive input/output dead times as in Eq (11).
    Section 2.5 and Eq (12); the success of the DTS method depends on the true delay matrix being close to the additive form theta_i + tau_j.
  • domain assumption The per-path dead time matrix delta is known or estimated accurately before solving the LP.
    Section 3.4 step 1; if delta is biased, the constraints Fx <= vec(delta) are wrong and the rectification in Eq (16) is not causal.
  • domain assumption The randomized SVD plus LOO estimator provides a reliable proxy for the ROM error; the dropped term in Eq (22) is negligible.
    Section 3.5; the adaptive pipeline stops based on gamma_LOO, which is explicitly heuristic and not proven to bound the true error.

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Cite this review

Pith. "Pith review of Adaptive Reduced Order Modelling of Discrete-Time Systems with Input-Output Dead Time." pith.science (2026). https://pith.science/paper/LMDNOAVB

@misc{pith2026250608870,
  author       = {Pith},
  title        = {Pith review of: Adaptive Reduced Order Modelling of Discrete-Time Systems with Input-Output Dead Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMDNOAVB}},
  note         = {Machine review of arXiv:2506.08870}
}
read the original abstract

While many acoustic systems are well-modelled by linear time-invariant dynamical systems, high-fidelity models often become computationally expensive due the complexity of dynamics. Reduced order modelling techniques, such as the Eigensystem Realization Algorithm (ERA), can be used to create efficient surrogate models from measurement data, particularly impulse responses. However, practical challenges remain, including the presence of input-output dead times, i.e. propagation delays, in the data, which can increase model order and introduce artifacts like pre-ringing. This paper introduces an improved technique for the extraction of dead times, by formulating a linear program to separate input and output dead times from the data. Additionally, the paper presents an adaptive randomized ERA pipeline that leverages recent advances in numerical linear algebra to reduce computational complexity and enabling scalable model reduction. Benchmarking on large-scale datasets of measured room impulse responses demonstrates that the propsed dead time extraction scheme yields more accurate and efficient reduced order models compared to previous approaches. The implementation is made available as open-source Python code, facilitating reproducibility and further research.

Figures

Figures reproduced from arXiv: 2506.08870 by the authors.

Figure 1
Figure 1. Schematic representation of different LTI model structures for a MIMO system [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Frequency responses of different ROMs for the MIRACLE-D1 scenario. Each [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Schematic representation of different IO dead time LTI model structures for a [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: The dead time splitting problem for a simple acoustic free field transmission [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Geometrical setup of the MIRACLE scenarios. The sources (positionable loud [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Geometrical setup of the MIRD scenarios. The 26 sources are equiangularly [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Comparison of relative error, bounds, and estimators. The figure depicts the [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Comparison of dead time extraction methods for selected benchmark scenarios. [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: Performance overview of randomized ERA and the proposed LOO error esti [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]
Figure 10
Figure 10. Figure 10: IRs of different ROMs for the MIRACLE-D1 scenario. Each subplot depicts [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]
Figure 11
Figure 11. Figure 11: Frequency responses of the largest constructed ROMs. Each subplot depicts [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: Overview of the computational runtimes.. The plot depicts the runtime in [PITH_FULL_IMAGE:figures/full_fig_p031_12.png]

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Forward citations

Cited by 1 Pith paper

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