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arxiv: 2601.08372 · v2 · submitted 2026-01-13 · 📡 eess.SY · cs.SY· math.OC

Data-Driven Regularized Time-Limited h2 Model Reduction from Noisy Impulse Responses

Hiroki Sakamoto , Kazuhiro Sato This is my paper

Pith reviewed 2026-05-16 15:18 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords model reductiontime-limited H2data-drivennoisy dataimpulse responsesregularizationdiscrete-time systemssystem identification
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The pith

Regularization enables accurate time-limited H2 model reduction from noisy impulse response data alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a data-driven method for reducing the order of discrete-time linear systems to match their impulse responses over a finite time window in the H2 sense. The approach adds regularization to the optimization problem and shows that both the cost and its gradient can be computed using only the noisy impulse responses. On standard benchmark systems, the regularized version produces lower relative errors than unregularized and other methods, particularly when noise would otherwise degrade performance. The result matters because many real systems provide only noisy measurements rather than clean models or data.

Core claim

The authors formulate and solve a regularized time-limited H2 model reduction problem using only noisy impulse response data for discrete-time LTI systems, demonstrating that the objective function and gradient can be represented solely from this data, leading to better accuracy than tested alternatives under noise.

What carries the argument

The regularized time-limited H2 objective, whose value and gradient are computed directly from noisy impulse responses without needing the full system matrices.

If this is right

  • The method yields lower relative time-limited H2 errors than unregularized approaches on SLICOT benchmarks.
  • It remains effective when noise causes the unregularized method to produce worse approximations.
  • The optimization can be performed without knowledge of exact system order or noise statistics beyond the data.
  • Both the objective and gradient depend only on the available noisy impulse responses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may generalize to other model reduction norms or continuous-time systems if similar data representations exist.
  • Automatic selection rules for the regularization parameter could further reduce the need for tuning.
  • Real-time or online model reduction from streaming noisy data becomes more feasible.

Load-bearing premise

That an appropriate regularization parameter can be selected to improve results without prior knowledge of the exact noise level or true system order.

What would settle it

An experiment on a noisy impulse response dataset from a benchmark system where the regularized method's relative H2 error is higher than the unregularized method for any reasonable parameter choice.

Figures

Figures reproduced from arXiv: 2601.08372 by Hiroki Sakamoto, Kazuhiro Sato.

Figure 1
Figure 1. Figure 1: Overview of the proposed data-driven time-limited [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the convergence behavior of Algorithm 1 for L = 20. For all cases, the relative error ∥H−Hˆ ∥h2 L ∥H∥h2 L is improved compared with the ERA-based initialization. Moreover, when the noise standard deviation σ is small, the convergence behavior is almost identical to the noise-free case, whereas a larger σ degrades the convergence [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence behavior of Algorithm 1 (L = 40) initialized by ERA under three noise levels: noise-free, σ = 1, and σ = 50. ACKNOWLEDGMENT This work was supported by JSPS KAKENHI under Grant Numbers 23K28369 and 25KJ0986. APPENDIX A. PROOF Proof of Proposition 1. Since ⟨X, Y ⟩ = tr(X⊤Y ), the objec￾tive function can be written as fL( ˆθ) = ⟨Cˆ⊤C, P ˆ L⟩ − 2⟨C ⊤C, R ˆ L⟩. Hence dfL = 2⟨CPˆ L − CRL, dCˆ⟩ + ⟨Cˆ⊤… view at source ↗
read the original abstract

This paper develops a data-driven time-limited h2 model reduction method for discrete-time linear time-invariant systems. Specifically, we formulate and solve a regularized time-limited h2 model reduction problem using only noisy impulse response data. Furthermore, we show that the objective function and its gradient can be represented using only noisy impulse response data. Numerical experiments using SLICOT benchmarks demonstrate that the proposed regularized method achieves lower relative time-limited h2 errors than the tested alternatives and is effective in situations where the unregularized method may deteriorate under noise.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper develops a data-driven regularized time-limited H2 model reduction method for discrete-time LTI systems. It formulates a regularized optimization problem solved using only noisy impulse response data, derives expressions for the objective function and its gradient directly from this data, and reports numerical experiments on SLICOT benchmarks showing lower relative time-limited H2 errors than unregularized and alternative methods, particularly when noise causes the unregularized approach to deteriorate.

Significance. If the central empirical claims hold, the work offers a practical extension of data-driven model reduction techniques by incorporating regularization to improve robustness to noise without requiring system identification or clean data. The direct use of impulse responses for both objective and gradient is a methodological strength that avoids intermediate fitting steps.

major comments (2)
  1. [Numerical experiments] Numerical experiments section: the regularization parameter is selected via grid search or heuristic on the SLICOT examples, but no procedure is given that operates solely from the noisy impulse responses without knowledge of noise covariance or true system order; this undermines the claim that the method is effective from noisy data alone, as the selection implicitly requires validation against unavailable ground truth.
  2. [§3] Derivation of objective and gradient (abstract and §3): the manuscript states that the objective and gradient are expressed using only noisy impulse response data, yet provides no explicit equations, error bounds, or analysis of how noise propagates into the time-limited H2 norm approximation; without this, the support for the central data-driven claim remains incomplete.
minor comments (1)
  1. [Abstract] The abstract could more precisely state the noise levels and number of benchmarks used in the tests to allow readers to assess the scope of the reported improvements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the changes we will make in the revised version.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: the regularization parameter is selected via grid search or heuristic on the SLICOT examples, but no procedure is given that operates solely from the noisy impulse responses without knowledge of noise covariance or true system order; this undermines the claim that the method is effective from noisy data alone, as the selection implicitly requires validation against unavailable ground truth.

    Authors: We agree that the parameter selection in the current numerical experiments relies on knowledge of the true system, which is unavailable in practice and weakens the claim of operating from noisy data alone. In the revision we will add a practical data-driven procedure for choosing the regularization parameter using only the noisy impulse responses (e.g., an L-curve criterion based on the residual of the regularized objective or a discrepancy principle when a noise-level estimate is available). We will also report results obtained with this procedure on the SLICOT benchmarks to demonstrate that the method remains effective without ground-truth information. revision: yes

  2. Referee: [§3] Derivation of objective and gradient (abstract and §3): the manuscript states that the objective and gradient are expressed using only noisy impulse response data, yet provides no explicit equations, error bounds, or analysis of how noise propagates into the time-limited H2 norm approximation; without this, the support for the central data-driven claim remains incomplete.

    Authors: The derivations expressing the objective and gradient directly from the (noisy) impulse-response matrices are already present in Section 3, but we acknowledge that they are not stated with sufficient explicitness and that noise-propagation analysis is absent. In the revision we will (i) display the explicit closed-form expressions for both the objective and its gradient in terms of the noisy data, (ii) add a short derivation outline, and (iii) include a brief analysis of how bounded additive noise in the impulse responses propagates into the computed time-limited H2 norm together with first-order error bounds under standard noise assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation; data-driven formulation is self-contained

full rationale

The paper formulates the regularized time-limited H2 objective directly from noisy impulse responses and shows explicit representations for the objective and gradient using only that data. No step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. Numerical validation on SLICOT benchmarks provides independent empirical support rather than tautological equivalence to inputs. The regularization parameter choice is presented as a practical tuning step without load-bearing self-referential theorems.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard assumption that the underlying system is discrete-time linear time-invariant and introduces a tunable regularization parameter whose value is chosen to balance fit and smoothness.

free parameters (1)
  • regularization parameter
    Tuned during numerical experiments to achieve lower errors under noise; its selection is not derived from first principles.
axioms (1)
  • domain assumption The system is discrete-time linear time-invariant
    Invoked throughout the formulation of the time-limited H2 problem.

pith-pipeline@v0.9.0 · 5387 in / 1066 out tokens · 41617 ms · 2026-05-16T15:18:54.449003+00:00 · methodology

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