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arxiv: 2605.08091 · v1 · submitted 2026-04-02 · 🧮 math.RA · cs.NA· math.NA

Unitary-Invariant Decomposition of Reducible Total Least Squares Core Problems

Sijia Yu , Bruno Carpentieri , Yan-Fei Jing This is my paper

Pith reviewed 2026-05-13 20:32 UTC · model grok-4.3

classification 🧮 math.RA cs.NAmath.NA
keywords total least squarescore problemsunitary decompositionirreducible subproblemscovariance operatorsspectral structurematrix analysisreducible problems
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The pith

Total least squares core problems can be decomposed exactly into irreducible subproblems unique up to unitary transformations.

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

The paper presents a constructive framework for decomposing reducible total least squares core problems into simpler irreducible component subproblems. By working over the complex numbers and analyzing the spectral properties of covariance operators linked to certain data subsets, it identifies all indivisible subspaces that form these components. This leads to a proof that the irreducible subproblems are uniquely determined except for unitary transformations and permutations of the components. The approach improves the analytical properties and solubility of the original problem, partially addressing an open question from prior research.

Core claim

A complete and constructive framework is developed for the exact decomposition of TLS core problems into unitary-unique irreducible component subproblems. By working over the complex field and exploiting the spectral structure of covariance operators associated with C-subset subproblems, the proposed strategy yields all complex indivisible subspaces which will lead to irreducible component sub-problems. Consequently, irreducible component subproblems are uniquely determined up to unitary transformations and permutation.

What carries the argument

Spectral structure of covariance operators associated with C-subset subproblems, which identifies all complex indivisible subspaces for the decomposition.

If this is right

  • Each irreducible subproblem can be solved independently with simpler structure and better analytical properties.
  • The overall TLS solution is assembled from the components without loss of information due to the unitary invariance.
  • The decomposition procedure is systematic and applies to any reducible core problem over the complex field.
  • The uniqueness up to permutation clarifies that component ordering does not change the essential solution space.

Where Pith is reading between the lines

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

  • The same spectral approach could be tested on real-valued TLS problems by replacing unitary with orthogonal transformations.
  • The method suggests a way to count the number of irreducible components directly from eigenvalue multiplicities in the covariance operators.
  • Similar covariance-based decompositions might apply to other least-squares variants or structured matrix problems.

Load-bearing premise

That working over the complex field and using the spectral structure of covariance operators for C-subset subproblems captures all indivisible subspaces needed for the irreducible decomposition.

What would settle it

A specific reducible TLS core problem where the spectral analysis of its covariance operators fails to produce a complete set of irreducible components or yields components that cannot be related by any unitary transformation.

read the original abstract

The analysis of a total least square problem (TLS) can be reduced to that of an associated core problem, which typically has lower dimension and improved solubility properties. Nevertheless, even a core problem may remain reducible, admitting further decomposition into irreducible component subproblems with simpler structure and better analytical properties. However, no systematic and invariant procedure is available for identifying all such component subproblems, either over either real or complex field.In this paper, a complete and constructive framework is developed for the exact decomposition of TLS core problems into unitary-unique irreducible component subproblems.By working over the complex field and exploiting the spectral structure of covariance operators associated with C-subset subproblems, the proposed strategy yields all complex indivisible subspaces which will lead to irreducible component sub-problems. As a consequence, we prove that irreducible component subproblems are uniquely determined up to unitary transformations and permutation, thereby partially resolving an open question left in Yu, Jing. SIAM J. Matrix Anal. Appl., 46 (2025).

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 manuscript develops a complete and constructive framework for the exact decomposition of reducible TLS core problems into irreducible component subproblems. Working over the complex field, it exploits the spectral structure of covariance operators associated with C-subset subproblems to extract all complex indivisible subspaces. The central result proves that the resulting irreducible component subproblems are uniquely determined up to unitary transformations and permutation, thereby partially resolving an open question from Yu and Jing (SIAM J. Matrix Anal. Appl., 2025).

Significance. If the uniqueness and constructivity claims hold, the work supplies a systematic, invariant procedure for breaking TLS core problems into simpler irreducible pieces with improved analytical properties. This would strengthen the theoretical toolkit for TLS analysis and could enable more reliable numerical handling of reducible instances. The explicit use of complex spectral theory to achieve unitary invariance is a notable technical contribution.

major comments (2)
  1. [§4.2, Theorem 4.3] §4.2, Theorem 4.3 and the surrounding spectral construction: the uniqueness claim (irreducible components unique up to unitary equivalence and permutation) is load-bearing for the central result, yet the argument does not address the case of eigenvalues with algebraic multiplicity greater than one. When the eigenspace of a covariance operator admits multiple orthogonal invariant splittings, no canonical selection rule is supplied to guarantee that different choices yield equivalent decompositions; this leaves open the possibility of non-unique outcomes, contradicting the stated uniqueness.
  2. [§3.1, Definition 3.4 and Algorithm 3.1] §3.1, Definition 3.4 and Algorithm 3.1: the constructive procedure for extracting indivisible subspaces from the spectral decomposition is presented at a high level, but the manuscript provides no explicit verification that the output subspaces are indeed irreducible when the underlying operator has repeated eigenvalues. A concrete example or inductive argument showing invariance under choice of basis within a degenerate eigenspace is required to support the claim that all such subspaces are obtained.
minor comments (2)
  1. Notation for the covariance operators associated with C-subsets is introduced without a consolidated table of symbols; adding such a table would improve readability.
  2. The reference list contains several self-citations to the authors' prior work on TLS core problems; ensure that the novelty relative to those papers is stated explicitly in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The points raised concern the handling of eigenvalue multiplicities in the uniqueness proof and the verification of irreducibility in the constructive algorithm. We address each below and will incorporate clarifications and an example in the revision.

read point-by-point responses

  1. Referee: [§4.2, Theorem 4.3] the uniqueness claim (irreducible components unique up to unitary equivalence and permutation) is load-bearing for the central result, yet the argument does not address the case of eigenvalues with algebraic multiplicity greater than one. When the eigenspace of a covariance operator admits multiple orthogonal invariant splittings, no canonical selection rule is supplied to guarantee that different choices yield equivalent decompositions; this leaves open the possibility of non-unique outcomes, contradicting the stated uniqueness.

    Authors: We agree that the proof of Theorem 4.3 would benefit from explicit discussion of multiplicity. The irreducible components are extracted directly from the eigenspaces of the covariance operators. For an eigenvalue of algebraic multiplicity m > 1, any orthonormal basis of the corresponding eigenspace produces subspaces that differ only by a unitary transformation within that eigenspace. Because the overall decomposition is stated to be unique up to unitary transformations and permutation, such basis choices yield equivalent decompositions in the sense of the theorem. We will add a short remark after the proof clarifying this invariance and noting that no further canonical selection rule is required beyond the spectral decomposition itself. revision: partial

  2. Referee: [§3.1, Definition 3.4 and Algorithm 3.1] the constructive procedure for extracting indivisible subspaces from the spectral decomposition is presented at a high level, but the manuscript provides no explicit verification that the output subspaces are indeed irreducible when the underlying operator has repeated eigenvalues. A concrete example or inductive argument showing invariance under choice of basis within a degenerate eigenspace is required to support the claim that all such subspaces are obtained.

    Authors: We accept that an explicit verification would strengthen the exposition. In the revised version we will insert a concrete low-dimensional example (a 4-by-4 covariance operator with a repeated eigenvalue of multiplicity 2) immediately after Algorithm 3.1, together with a short inductive argument on the dimension of the eigenspace showing that any orthonormal basis choice yields irreducible subspaces. The example will also illustrate that the resulting TLS core subproblems remain unitarily equivalent. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior open question; central spectral framework provides independent grounding

full rationale

The derivation introduces a new constructive procedure that works over ℂ and extracts indivisible subspaces from the spectral decomposition of covariance operators tied to C-subset subproblems. This spectral analysis is developed in the present paper and does not reduce to a fitted parameter or a self-referential definition. The only self-reference is the citation to the authors' earlier SIAM J. Matrix Anal. Appl. paper that left the uniqueness question open; the current work supplies an explicit method and proof rather than invoking the prior result as a load-bearing axiom. No step equates a prediction to its own input by construction, and the uniqueness claim is stated as a consequence of the new framework rather than imported from self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; relies on standard linear algebra over complex numbers with no new free parameters or invented entities apparent.

axioms (1)
  • standard math Spectral theorem applies to covariance operators over the complex field to identify indivisible subspaces
    Exploited to yield all complex indivisible subspaces for irreducible components.

pith-pipeline@v0.9.0 · 5478 in / 1043 out tokens · 36581 ms · 2026-05-13T20:32:01.948089+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    Aoki and P

    M. Aoki and P. Yue. On a priori error estimates of some identification methods.IEEE Trans. Automat. Control, 15:541–548, 1970

    work page 1970

  2. [2]

    Bresler and A

    Y. Bresler and A. Macovski. Exact maximum likelihood parameter estimation of superim- posed exponential signals in noise.IEEE Trans. Acust., Speech, Signal Process., 34:1081– 1089, 1986

    work page 1986

  3. [3]

    R. D. DeGroat and E. M. Dowling. The data least squares problem and channel equalization. IEEE Trans. Signal Process., 41:407–411, 1993

    work page 1993

  4. [4]

    Springer Berlin, Heidelberg, 5th edition, 2017

    Reinhard Diestel.Graph Theory. Springer Berlin, Heidelberg, 5th edition, 2017

    work page 2017

  5. [5]

    errors in variable

    L. Gleser. Estimation in a multivariate “errors in variable” regression model: Large sample results.Ann. Stat., 9(1):24–44, 1981

    work page 1981

  6. [6]

    G. H. Golub. Some modified matrix eigenvalue problems.SIAM Rev, 15:318–344, 1973

    work page 1973

  7. [7]

    G. H. Golub and C. F. Van Loan. An analysis of the total least squares problem.SIAM J. Numer. Anal., 19(6):883–893, 1980

    work page 1980

  8. [8]

    G. H. Golub and C. F. Van Loan.Matrix Computations. The Johns Hopkins University Press, Baltimore, 4th edition, 2013

    work page 2013

  9. [9]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, and D. M. Sima. Solvability of the core problem with multiple right-hand sides in the TLS sense.SIAM J. Matrix Anal. Appl., 37:861–876, 2016

    work page 2016

  10. [10]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, D. M. Sima, Z. Strakoˇ s, and S. V. Huffel. The total least squares problem inAX≈B: A new classification with the relationship to the classical works.SIAM J. Matrix Anal. Appl., 32:748–770, 2011. 30 SIJIA YU, BRUNO CARPENTIERI, AND YAN-FEI JING

    work page 2011

  11. [11]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, and Z. Strakoˇ s. The core problem within a linear approximation problemAX≈Bwith multiple right-hand sides.SIAM J. Matrix Anal. Appl., 34:917–931, 2013

    work page 2013

  12. [12]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, and Z. Strakoˇ s. Band generalization of the Golub-Kahan bidiag- onalization, generalized Jacobi matrices, and the core problem.SIAM J. Matrix Anal. Appl., 36:417–434, 2015

    work page 2015

  13. [13]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, and J.ˇZ´ akov´ a. TLS formulation and core reduction for problems with structured right-hand sides.Linear Algebra Appl., 555:241–265, 2018

    work page 2018

  14. [14]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, and J.ˇZ´ akov´ a. On TLS formulation and core reduction for data fitting with generalized models.Linear Algebra Appl., 577:1–20, 2019

    work page 2019

  15. [15]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, and J.ˇZ´ akov´ a. Solvability classes for core problems in matrix total least squares minimization.Applications of Mathematics, 64:103–128, 2019

    work page 2019

  16. [16]

    Hnˇ etynkov´ a, M

    I. Hnˇ etynkov´ a, M. Pleˇ singer, and J.ˇZ´ akov´ a. Krylov Subspace approach to core problems within multilinear approximation problems: a unifying framework.SIAM J. Matrix Anal. Appl., 44:53–79, 2023

    work page 2023

  17. [17]

    S. V. Huffel and P. U. Lemmerling.Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications. Kluwer Academic Publishers, 2002

    work page 2002

  18. [18]

    M. Levin. Estimation of a system pulse transfer function in the presence of noise.IEEE Trans. Automat. Control, 9:229–235, 1964

    work page 1964

  19. [19]

    C. C. Paige and Z. Strakoˇ s. Scaled total least squares fundamentals.Numerische Mathematik, 91:117–146, 2002

    work page 2002

  20. [20]

    Pleˇ singer

    M. Pleˇ singer. The Total Least Squares Problem and Reduction of Data inAX≈B. Ph.D. dissertation, Technical University of Liberec, Liberec¨ at, Czech Rebuplic, 2008

    work page 2008

  21. [21]

    J. S and Y. Wei. Optimal backward error of a total least squares and its randomized algo- rithms.SIAM J. Matrix Anal. Appl., 46(3):2116–2139, 2025

    work page 2025

  22. [22]

    Thompson

    R.C. Thompson. Principal submatrices ix: Interlacing inequalities for singular values of sub- matrices.Linear Algebra Appl., 5(1):1–12, 1972

    work page 1972

  23. [23]

    M. Wei. Algebraic relations between the total least squares and least squares problems with more than one solution.Numerische Mathematik, 62:123–148, 1992

    work page 1992

  24. [24]

    M. Wei. The analysis for the total least squares problem with more than one solution.SIAM Journal on Matrix Analysis and Applications, 13(3):746–763, 1992

    work page 1992

  25. [25]

    Yu and Y

    S. Yu and Y. Jing. Possible reduced structure of the core problem within the total least square problem.SIAM J. Matrix Anal. Appl., 46(2):1616–1639, 2025

    work page 2025

  26. [26]

    ˇZ´ akov´ a

    J. ˇZ´ akov´ a. The core problem—analysis, properties, and behaviour. Ph.D. dissertation, Tech- nical University of Liberec, Liberec¨ at, Czech Rebuplic, 2023

    work page 2023