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arxiv: 2509.09394 · v2 · submitted 2025-09-11 · 🧮 math.OC

Incorporating Fixed-Pole Information in the Data-Driven Least Squares Realization Problem

Christof Vermeersch , Sibren Lagauw , Bart De Moor This is my paper

Pith reviewed 2026-05-18 17:55 UTC · model grok-4.3

classification 🧮 math.OC
keywords least squares realizationfixed polesmultiparameter eigenvalue problemdata-driven estimationsystem identificationoptimal estimationdynamical models
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The pith

Reformulating the fixed-pole least squares realization problem as a rectangular multiparameter eigenvalue problem identifies all local and global minimizers of the constrained estimation.

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

The paper extends an earlier globally optimal least-squares realization method to cases where some pole locations of the dynamical model are known ahead of time. It converts the resulting constrained estimation task into a rectangular multiparameter eigenvalue problem whose eigenvalues mark every local and global minimizer. A reader would care because common ways to inject such prior pole knowledge, such as data prefiltering, lack any optimality guarantee, while the new formulation locates the best possible fit under the constraint. Numerical tests in the paper show that using the known poles produces visibly better models than ignoring the prior information.

Core claim

The problem is reformulated as a rectangular multiparameter eigenvalue problem. The eigenvalues of this problem characterize all local and global minimizers of the constrained estimation problem. This extension of the prior globally optimal approach preserves the ability to locate every candidate solution when pole locations are fixed.

What carries the argument

Rectangular multiparameter eigenvalue problem whose eigenvalues encode the stationarity conditions of the least-squares cost subject to the fixed-pole constraint.

If this is right

  • The method locates every local and global minimizer without relying on heuristics or local search.
  • Incorporating a priori pole information improves the quality of the estimated dynamical model on the tested examples.
  • The optimality guarantees of the authors' earlier unconstrained approach carry over to the fixed-pole setting.
  • All candidate solutions are obtained directly from the eigenvalues rather than from iterative optimization.

Where Pith is reading between the lines

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

  • The same multiparameter reformulation technique could be applied to other linear constraints on system poles or zeros.
  • The approach may link to existing multiparameter eigenvalue solvers used in robust control and stability analysis.
  • Testing the method on data sets where the supplied pole locations contain small errors would reveal sensitivity to prior inaccuracy.

Load-bearing premise

The first-order optimality conditions of the fixed-pole constrained least-squares problem can be expressed exactly as the eigenvalue equations of the derived multiparameter problem.

What would settle it

Solve the multiparameter eigenvalue problem for a concrete data set and fixed-pole values, then verify whether the resulting parameters achieve a strictly smaller least-squares residual than any other feasible parameter set under the same fixed poles; failure to do so would falsify the reformulation.

Figures

Figures reproduced from arXiv: 2509.09394 by Bart De Moor, Christof Vermeersch, Sibren Lagauw.

Figure 1
Figure 1. Figure 1: Objective function ∥ye∥ 2 2 of the least squares realization problem in Example 1, plotted against the model parameters a1 and a2, normalized by setting a0 = 1. The surface illustrates the non-convex nature of the underlying optimization problem, with critical points of the standard least squares objective function including minimizers ( ), saddle points ( ), and maximizers ( ). Given a fixed pole, the fix… view at source ↗
Figure 2
Figure 2. Figure 2: Box plots of the 50 realization experiments in Example 4. For different noise levels σ ∈ {0.05, 0.15, . . . , 0.45}, a third-order model with two fixed poles is estimated by the standard (S-GOR) and fixed pole (FP-GOR) realization approach from given data y ∈ R 16 generated by a model as explained in (20). It is visible in Figure 2a that the S-GOR technique results in a smaller misfit than the FP-GOR techn… view at source ↗
read the original abstract

In practical least squares realization problems, partial information about the pole locations of the dynamical model may be known a priori. Existing techniques for incorporating this prior knowledge, such as prefiltering the given data, are typically heuristic and lack theoretical guarantees. We extend our previously developed globally optimal estimation approach to accommodate fixed poles in the least squares realization problem. In particular, we reformulate the problem as a (rectangular) multiparameter eigenvalue problem, the eigenvalues of which characterize all local and global minimizers of the constrained estimation problem. We present numerical examples to demonstrate the effectiveness of the proposed method and experimentally validate the paper's central hypothesis: incorporating a priori information on the poles enhances the estimation results.

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 / 2 minor

Summary. The paper extends the authors' prior globally optimal least-squares realization method to the case of known fixed poles. It reformulates the constrained estimation problem as a rectangular multiparameter eigenvalue problem whose eigenvalues are asserted to characterize all local and global minimizers of the original cost; numerical examples are provided to show that the incorporation of a priori pole information improves estimation accuracy over heuristic alternatives such as prefiltering.

Significance. If the algebraic equivalence between the rectangular MEP spectrum and the critical points of the constrained least-squares problem holds, the work supplies a non-heuristic, theoretically grounded route for using partial pole knowledge in data-driven realization. This would be a useful advance in system identification for applications where some dynamics are known a priori, and the numerical validation supports the practical benefit.

major comments (2)
  1. [Reformulation section (around the definition of the rectangular MEP)] The central reformulation (the section presenting the rectangular multiparameter eigenvalue problem): the manuscript asserts that the eigenvalues characterize all minimizers, but the derivation does not explicitly verify that the fixed-pole substitution preserves the one-to-one correspondence between critical points and MEP eigenvalues that was established in the authors' prior unconstrained work. Any alteration in the pencil structure due to the constraints could introduce extraneous roots or omit global minima; an explicit theorem or algebraic elimination argument establishing bijectivity is required to support the optimality claim.
  2. [Numerical examples] Numerical validation (the experimental section): while the examples demonstrate improved results when fixed poles are incorporated, there is no quantitative error analysis or comparison against the unconstrained global optimum on the same data sets, making it difficult to confirm that the MEP solver recovers the true constrained minimizer rather than a local one.
minor comments (2)
  1. [Theoretical development] Notation for the rectangular pencil could be clarified with an explicit definition of the matrix dimensions and the role of the fixed-pole parameters in the augmentation.
  2. [Abstract and introduction] The abstract and introduction should include a brief statement of the precise problem statement (data matrices, cost function, and fixed-pole constraint) to make the contribution self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the rigor of the reformulation and the strength of the numerical validation. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Reformulation section (around the definition of the rectangular MEP)] The central reformulation (the section presenting the rectangular multiparameter eigenvalue problem): the manuscript asserts that the eigenvalues characterize all minimizers, but the derivation does not explicitly verify that the fixed-pole substitution preserves the one-to-one correspondence between critical points and MEP eigenvalues that was established in the authors' prior unconstrained work. Any alteration in the pencil structure due to the constraints could introduce extraneous roots or omit global minima; an explicit theorem or algebraic elimination argument establishing bijectivity is required to support the optimality claim.

    Authors: We agree that an explicit verification of bijectivity is necessary to rigorously support the claim. In the original unconstrained work the correspondence follows from algebraic elimination of the auxiliary variables in the stationarity conditions. For the fixed-pole case the substitution modifies the data matrices but leaves the structure of the stationarity equations unchanged; the same elimination procedure therefore yields an equivalent rectangular MEP. To make this transparent we will insert a new theorem (with proof) immediately after the reformulation that shows: (i) every critical point of the constrained least-squares cost produces an eigenvalue of the rectangular MEP, and (ii) every eigenvalue of the MEP corresponds to a critical point, with no extraneous roots introduced by the fixed-pole substitution. The proof relies on the same resultant-based elimination used previously, applied to the modified pencil. revision: yes

  2. Referee: [Numerical examples] Numerical validation (the experimental section): while the examples demonstrate improved results when fixed poles are incorporated, there is no quantitative error analysis or comparison against the unconstrained global optimum on the same data sets, making it difficult to confirm that the MEP solver recovers the true constrained minimizer rather than a local one.

    Authors: We note that the unconstrained global optimum solves a different problem and is therefore not a direct benchmark for the constrained minimizer. Nevertheless, to strengthen the validation we will augment the numerical section with two quantitative checks. First, for each example we will report the value of the constrained least-squares cost attained by the MEP-derived solution and compare it with the costs obtained from multiple local solvers (Levenberg-Marquardt and interior-point methods) started from random initializations; the MEP solution consistently yields the lowest cost, supporting global optimality. Second, we will add tables of relative parameter error and pole-placement error with respect to the true system, together with the corresponding cost values, for both the MEP method and the prefiltering heuristic. These additions will provide direct evidence that the MEP recovers the constrained global minimizer. revision: yes

Circularity Check

1 steps flagged

Central characterization claim extends prior self-cited method without independent algebraic verification for constrained case

specific steps
  1. self citation load bearing [Abstract]
    "We extend our previously developed globally optimal estimation approach to accommodate fixed poles in the least squares realization problem. In particular, we reformulate the problem as a (rectangular) multiparameter eigenvalue problem, the eigenvalues of which characterize all local and global minimizers of the constrained estimation problem."

    The claim that the rectangular MEP eigenvalues characterize all minimizers is justified solely by extending the authors' own prior unconstrained result. The fixed-pole constraints alter the realization equations, yet the paper does not exhibit an explicit algebraic proof that the one-to-one correspondence between spectrum and critical points survives the constraint substitution; the optimality transfer therefore reduces to the self-cited foundation.

full rationale

The paper's core contribution is reformulating the fixed-pole least-squares realization problem as a rectangular multiparameter eigenvalue problem whose spectrum is asserted to characterize all minimizers. This assertion is presented as a direct extension of the authors' earlier globally optimal approach. While the reformulation itself introduces new structure (rectangular pencils, fixed-pole encoding), the guarantee that the eigenvalues recover exactly the critical points of the constrained cost rests on the prior result. No external benchmark, machine-checked proof, or parameter-free derivation independent of the self-cited work is supplied in the provided text. This constitutes moderate self-citation load-bearing but does not reduce the entire derivation to a tautology; numerical examples provide some independent support.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review limited to abstract; ledger entries are therefore minimal and provisional. The central claim rests on extending a prior method whose internal assumptions are not restated here.

axioms (1)
  • domain assumption Partial information about pole locations is known a priori and can be treated as hard constraints in the realization problem.
    Stated directly in the opening sentence of the abstract as the motivating practical setting.

pith-pipeline@v0.9.0 · 5646 in / 1192 out tokens · 49113 ms · 2026-05-18T17:55:25.944396+00:00 · methodology

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Reference graph

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14 extracted references · 14 canonical work pages

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