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arxiv: 2605.00585 · v1 · submitted 2026-05-01 · 📡 eess.SP · cs.NA· math.NA

Local Geometry of Least Squares for Unmixing Signals with Parameter-Dependent Dictionaries

Santos Michelena , Maxime Ferreira Da Costa , Jos\'e Picheral This is my paper

Pith reviewed 2026-05-09 19:03 UTC · model grok-4.3

classification 📡 eess.SP cs.NAmath.NA
keywords unmixing metricvariable projectionseparable nonlinear least squaresparametric coherencepoint spread function unmixinglocal convergencesignal processingdictionary atoms
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The pith

The unmixing metric captures the distinct roles of linear and nonlinear parameters to deliver local convergence and stability in signal unmixing.

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

Signals are often modeled as sums of dictionary atoms whose shapes depend nonlinearly on unknown parameters. The paper introduces the unmixing metric to measure closeness while respecting that linear coefficients and nonlinear parameters play different roles and have different sensitivities. Under this metric the least-squares problem is shown to have local convergence and stable recovery. Variable projection is reinterpreted as a search that stays on the manifold of linear parameters that best fit any given nonlinear parameters, which accounts for its observed practical advantages and yields sharper bounds. The general theory is then applied to unmixing point-spread functions, where a parametric coherence quantity shows that the separation between signal supports directly sets the size of the region of convergence and the stability of the recovered parameters.

Core claim

The paper shows that for separable least-squares unmixing problems the unmixing metric induces a local topology in which convergence and stability hold. It further demonstrates that variable projection is equivalent to restricting the optimization to the manifold of optimal linear parameters for each fixed set of nonlinear parameters. This geometric restriction provides both a principled explanation for the better performance of variable projection and sharp theoretical guarantees. Specializing to point-spread-function unmixing, the authors define parametric coherence and prove that support separation controls the convergence region size and the recovery stability.

What carries the argument

The unmixing metric, which is a distance designed to reflect the separate sensitivities of the linear coefficients and the nonlinear parameters in the signal model.

Load-bearing premise

The dictionary atoms must depend smoothly on the nonlinear parameters so that a local topology can be defined with the unmixing metric and a nonempty convergence region exists when the supports are separated.

What would settle it

Running least-squares unmixing on two point-spread functions whose separation falls below the parametric-coherence threshold should produce either no convergence or convergence to an erroneous set of parameters.

Figures

Figures reproduced from arXiv: 2605.00585 by Jos\'e Picheral, Maxime Ferreira Da Costa, Santos Michelena.

Figure 1
Figure 1. Figure 1: Visualization of the optimization geometry of a separable problem. view at source ↗
Figure 2
Figure 2. Figure 2: Geometric effect of variable projection on the loss landscape. view at source ↗
Figure 3
Figure 3. Figure 3: Empirical validation of Theorem 3 for a Gaussian kernel. (a) view at source ↗
Figure 4
Figure 4. Figure 4: Coherence for kernels with different tail decays. (a) kernels view at source ↗
Figure 5
Figure 5. Figure 5: Noiseless empirical basin estimates for least squares. Blue: sampled view at source ↗
Figure 8
Figure 8. Figure 8: Noisy empirical basin estimates for projected least squares at view at source ↗
Figure 6
Figure 6. Figure 6: Noiseless empirical basin estimates for projected least squares. view at source ↗
Figure 9
Figure 9. Figure 9: Empirical recovery error and analytical stability bounds in the PSF view at source ↗
Figure 10
Figure 10. Figure 10: Empirical convergence behavior of Levenberg–Marquardt (blue tri view at source ↗
Figure 11
Figure 11. Figure 11: Empirical convergence radius and analytical strong convexity bounds for variable projection at view at source ↗
read the original abstract

Modeling signals as linear combinations of atoms from a dictionary is ubiquitous in modern signal processing. In the finite-dimensional setting, whenever atoms depend nonlinearly upon unknown parameters, the signal model is said to be separable. In this work, we study least-squares reconstruction of separable signals and establish a unified theoretical framework for their analysis. We introduce the unmixing metric, a distance that captures the distinct roles and sensitivities of linear and nonlinear parameters, and establish local convergence and stability guarantees under its topology. We then analyze variable projection from a geometric perspective, showing that it corresponds to restricting the optimization to the manifold of optimal linear parameters. This viewpoint provides a principled explanation for the improved algorithmic behavior of variable projection observed in practice, and produces sharp theoretical guarantees. The generic theory for separable problems is specialized to the case of point spread function (PSF) unmixing. We introduce a parametric notion of coherence and show that support separation directly controls both the size of the convergence region and the stability of recovery. Numerical experiments corroborate the theoretical predictions and demonstrate the practical relevance of the proposed framework.

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 develops a geometric framework for least-squares unmixing of separable signals whose dictionary atoms depend nonlinearly on parameters. It defines an unmixing metric that distinguishes the roles of linear and nonlinear parameters, proves local convergence and stability results under the induced topology, and shows that variable projection corresponds to restricting the search to the manifold of optimal linear coefficients. The general theory is specialized to point-spread-function (PSF) unmixing by introducing a parametric coherence quantity; the authors claim that minimum support separation directly governs both the radius of the convergence basin and the stability of recovery. Numerical experiments are presented to corroborate the predictions.

Significance. If the local-geometry claims hold, the work supplies a principled explanation for the observed superiority of variable projection over joint nonlinear least squares and yields sharp, topology-specific guarantees that are currently unavailable for separable problems. The introduction of the unmixing metric and the parametric-coherence notion are original contributions that could be useful beyond PSF unmixing. The manuscript is self-contained and does not rely on external benchmarks for its core definitions.

major comments (2)
  1. [§5] §5 (PSF specialization) and the paragraph following Eq. (parametric coherence definition): the claim that support separation 'directly controls' the size of the convergence region requires an explicit lower bound on separation that keeps parametric coherence below the threshold guaranteeing a nonempty open basin under the unmixing metric. No such quantitative bound is derived or verified against the separations used in the numerical examples of §6; if the bound is violated, the local strong-convexity argument and the variable-projection manifold interpretation cease to apply.
  2. [Theorem 3] Theorem 3 (local convergence under unmixing metric): the proof sketch assumes the signals are separable and the atoms are sufficiently smooth, yet the manuscript does not state the precise smoothness or injectivity conditions on the nonlinear parameter map that are needed for the metric to induce a valid local topology. Without these, it is unclear whether the stated guarantees extend to the PSF atoms employed in the experiments.
minor comments (2)
  1. [§3] Notation for the unmixing metric is introduced without an explicit comparison table to the Euclidean or parameter-wise distances; a short side-by-side definition would improve readability.
  2. [Figure 4] Figure 4 (recovery error vs. separation): the plotted curves lack error bars or multiple random initializations, making it difficult to assess variability of the observed basin sizes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address the major comments point by point below. The revisions will strengthen the clarity and applicability of the theoretical results without altering the core contributions.

read point-by-point responses

  1. Referee: [§5] §5 (PSF specialization) and the paragraph following Eq. (parametric coherence definition): the claim that support separation 'directly controls' the size of the convergence region requires an explicit lower bound on separation that keeps parametric coherence below the threshold guaranteeing a nonempty open basin under the unmixing metric. No such quantitative bound is derived or verified against the separations used in the numerical examples of §6; if the bound is violated, the local strong-convexity argument and the variable-projection manifold interpretation cease to apply.

    Authors: We agree that an explicit lower bound on minimum support separation would make the dependence precise and permit direct verification against the experiments. In the revised manuscript we will add a corollary (following the parametric coherence definition in §5) that supplies a sufficient separation threshold in terms of the unmixing-metric constants and the coherence bound guaranteeing a nonempty open basin. We will also confirm that the separations employed in §6 satisfy this threshold, thereby ensuring the local strong-convexity and variable-projection interpretations remain valid for the reported numerical results. revision: yes

  2. Referee: [Theorem 3] Theorem 3 (local convergence under unmixing metric): the proof sketch assumes the signals are separable and the atoms are sufficiently smooth, yet the manuscript does not state the precise smoothness or injectivity conditions on the nonlinear parameter map that are needed for the metric to induce a valid local topology. Without these, it is unclear whether the stated guarantees extend to the PSF atoms employed in the experiments.

    Authors: The proof of Theorem 3 relies on the nonlinear parameter map being C²-smooth and locally injective so that the unmixing metric induces a valid local topology. We will explicitly insert these hypotheses into the statement of Theorem 3. For the PSF atoms used in the experiments we will add a short remark noting that common parametric models (Gaussian, Airy, or Moffat profiles) are C^∞ and locally injective for distinct parameter values, thereby confirming that the local convergence guarantees apply to the numerical examples. revision: yes

Circularity Check

0 steps flagged

No circularity: new metric and coherence are definitional; derivation self-contained

full rationale

The paper introduces the unmixing metric and parametric coherence as original definitions, then derives local convergence, stability guarantees, and the geometric interpretation of variable projection directly from the separable signal model and smoothness assumptions. No load-bearing step reduces by construction to a fitted input, self-citation chain, or renamed known result; the PSF specialization follows from the general theory without circularity. The framework is independent of external benchmarks and does not rely on prior author work to force uniqueness or ansatz choices.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper rests on standard signal-processing modeling assumptions and introduces two new conceptual objects without additional fitted constants or external entities.

axioms (2)
  • domain assumption Signals are modeled as linear combinations of atoms from a parameter-dependent dictionary (separable model)
    Core modeling premise stated in the abstract for all subsequent analysis.
  • domain assumption Local neighborhoods exist around true parameters where the unmixing metric induces a suitable topology for convergence
    Invoked to establish local convergence and stability guarantees.
invented entities (2)
  • unmixing metric no independent evidence
    purpose: Distance that captures distinct roles and sensitivities of linear versus nonlinear parameters
    Newly defined object used to equip the parameter space with a topology for local analysis.
  • parametric notion of coherence no independent evidence
    purpose: Measure linking support separation to size of convergence region and stability in PSF unmixing
    Introduced specifically for the PSF specialization.

pith-pipeline@v0.9.0 · 5497 in / 1676 out tokens · 56027 ms · 2026-05-09T19:03:28.560810+00:00 · methodology

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    D. B. C. Salzer, M. I. Español, and G. Jeronimo,Variable projection methods for solving regularized separable inverse problems with applications to semi-blind image deblurring, 2026. arXiv: 2601.05224 [math.NA]

    work page arXiv 2026

  2. [2]

    Algorithms for separable nonlinear least squares with application to modelling time- resolved spectra

    K. M. Mullen, M. Vengris, and I. H. M. van Stokkum, “Algorithms for separable nonlinear least squares with application to modelling time- resolved spectra”,Journal of Global Optimization, vol. 38, pp. 201– 213, 2007

    work page 2007

  3. [3]

    Generalized rational variable projection with application in ecg compression

    P. Kovács, S. Fridli, and F. Schipp, “Generalized rational variable projection with application in ecg compression”,IEEE Transactions on Signal Processing, vol. 68, pp. 478–492, 2020

    work page 2020

  4. [4]

    Variable projection for computa- tional pdes with artificial neural networks

    Y . Wang, Y . Chen, and X. Zhang, “Variable projection for computa- tional pdes with artificial neural networks”, inProceedings of the 2024 Symposium on Scientific Computing and Machine Learning (SCML), 2024

    work page 2024

  5. [5]

    Nocedal and S

    J. Nocedal and S. J. Wright,Numerical Optimization, 2nd. New York, NY , USA: Springer, 2006

    work page 2006

  6. [6]

    L. V . Kantorovich and G. P. Akilov,Functional Analysis, 2nd ed. Oxford: Pergamon Press, 1982

    work page 1982

  7. [7]

    A kantorovich-type convergence analysis for the gauss-newton-method

    W. M. Häussler, “A kantorovich-type convergence analysis for the gauss-newton-method”,Numer. Math., vol. 48, no. 1, pp. 119–125, Jan. 1986

    work page 1986

  8. [8]

    O. P. Ferreira, M. L. N. Goncalves, and P. R. Oliveira,Local conver- gence analysis of Gauss-Newton’s method under majorant condition, arXiv:1003.5004 [math], Mar. 2010

    work page arXiv 2010

  9. [9]

    Kantorovich’s majorants principle for Newton’s method

    O. P. Ferreira and B. F. Svaiter, “Kantorovich’s majorants principle for Newton’s method”, en,Computational Optimization and Applications, vol. 42, no. 2, pp. 213–229, Mar. 2009

    work page 2009

  10. [10]

    Kantorovich’s theorem on newton’s method for solving generalized equations under the majorant condition

    G. N. Silva, “Kantorovich’s theorem on newton’s method for solving generalized equations under the majorant condition”,Applied Math- ematics and Computation, vol. 286, pp. 178–188, 2016

    work page 2016

  11. [11]

    The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate

    G. H. Golub and V . Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate”,SIAM Journal on Numerical Analysis, vol. 10, no. 2, pp. 413–432, 1973

    work page 1973

  12. [12]

    Separable nonlinear least squares: The variable projection method and its applications

    G. Golub and V . Pereyra, “Separable nonlinear least squares: The variable projection method and its applications”,Inverse Problems, vol. 19, no. 2, R1, Feb. 2003

    work page 2003

  13. [13]

    Algorithms for separable nonlinear least squares problems

    A. Ruhe and P. Å. Wedin, “Algorithms for separable nonlinear least squares problems”,SIAM Review, vol. 22, no. 3, pp. 318–337, 1980, Accessed: 2025-06-25

    work page 1980

  14. [14]

    Variable projection for nonlinear least squares problems

    D. O’Leary and B. Rust, “Variable projection for nonlinear least squares problems”,Computational Optimization and Applications, vol. 54, no. 3, pp. 579–593, Apr. 2013

    work page 2013

  15. [15]

    A variable projection method for solving separable non- linear least squares problems

    L. Kaufman, “A variable projection method for solving separable non- linear least squares problems”,BIT Numerical Mathematics, vol. 15, no. 1, pp. 49–57, 1975

    work page 1975

  16. [16]

    Multi-kernel unmixing and super- resolution using the modified matrix pencil method

    S. Chrétien and H. Tyagi, “Multi-kernel unmixing and super- resolution using the modified matrix pencil method”,Journal of Fourier Analysis and Applications, vol. 26, no. 1, p. 18, 2020

    work page 2020

  17. [17]

    Maximum likelihood for blind separation and deconvolution of noisy signals using mix- ture models

    E. Moulines, J.-F. Cardoso, and E. Gassiat, “Maximum likelihood for blind separation and deconvolution of noisy signals using mix- ture models”, in1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, ISSN: 1520-6149, vol. 5, Apr. 1997, 3617–3620 vol.5

    work page 1997

  18. [18]

    Identifiability in blind deconvolution with subspace or sparsity constraints

    Y . Li, K. Lee, and Y . Bresler, “Identifiability in blind deconvolution with subspace or sparsity constraints”,IEEE Transactions on infor- mation Theory, vol. 62, no. 7, pp. 4266–4275, 2016

    work page 2016

  19. [19]

    Multichannel sparse blind deconvolution on the sphere

    Y . Li and Y . Bresler, “Multichannel sparse blind deconvolution on the sphere”,IEEE Transactions on Information Theory, vol. 65, no. 11, pp. 7415–7436, 2019

    work page 2019

  20. [20]

    Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting

    R. Beinert and K. Bredies, “Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting”,Inverse Problems, vol. 35, no. 1, p. 015 002, Nov. 2018

    work page 2018

  21. [21]

    Compressed sensing with nonlinear observations and related nonlinear optimization problems

    T. Blumensath, “Compressed sensing with nonlinear observations and related nonlinear optimization problems”,IEEE Transactions on Information Theory, vol. 59, no. 6, pp. 3466–3474, 2013

    work page 2013

  22. [22]

    The basins of attraction of the global minimizers of the non-convex sparse spike estimation problem

    Y . Traonmilin and J.-F. Aujol, “The basins of attraction of the global minimizers of the non-convex sparse spike estimation problem”, Inverse Problems, vol. 36, no. 4, p. 045 003, 2020

    work page 2020

  23. [23]

    On strong basins of attractions for non-convex sparse spike estimation: Upper and lower bounds

    Y . Traonmilin, J.-F. Aujol, P.-J. Bénard, and A. Leclaire, “On strong basins of attractions for non-convex sparse spike estimation: Upper and lower bounds”,Journal of Mathematical Imaging and Vision, vol. 66, no. 1, pp. 57–74, 2024

    work page 2024

  24. [24]

    Local geometry of nonconvex spike deconvolution from low-pass measurements

    M. Ferreira Da Costa and Y . Chi, “Local geometry of nonconvex spike deconvolution from low-pass measurements”,IEEE Journal on Selected Areas in Information Theory, vol. 4, pp. 1–15, 2023

    work page 2023

  25. [25]

    Functional nonlinear sparse models

    L. F. O. Chamon, Y . C. Eldar, and A. Ribeiro, “Functional nonlinear sparse models”,IEEE Transactions on Signal Processing, vol. 68, pp. 2449–2463, 2020

    work page 2020

  26. [26]

    Three-dimensional super-resolution imaging by stochastic optical reconstruction mi- croscopy

    B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction mi- croscopy”,Science, vol. 319, no. 5864, pp. 810–813, 2008

    work page 2008

  27. [27]

    Inferring sparse representations of continuous signals with continuous orthog- onal matching pursuit

    K. C. Knudson, J. Yates, A. Huk, and J. W. Pillow, “Inferring sparse representations of continuous signals with continuous orthog- onal matching pursuit”,Advances in neural information processing systems, vol. 27, 2014

    work page 2014

  28. [28]

    Catching up on calibration-free LIBS

    F. Poggialini and colleagues, “Catching up on calibration-free LIBS”, Journal of Analytical Spectroscopy, vol. 8, pp. 1234–1245, 2023

    work page 2023

  29. [29]

    Convergence guarantees for unmixing PSFs over a manifold with non-convex optimization

    S. Michelena, M. Ferreira Da Costa, and J. Picheral, “Convergence guarantees for unmixing PSFs over a manifold with non-convex optimization”, in2025 IEEE Statistical Signal Processing Workshop (SSP), 2025, pp. 161–165

    work page 2025

  30. [30]

    Strong basin of attraction for unmixing kernels with the variable projection method

    S. Michelena, M. Ferreira Da Costa, and J. Picheral, “Strong basin of attraction for unmixing kernels with the variable projection method”, inICASSP 2026 - 2026 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2026, pp. 726–730

    work page 2026

  31. [31]

    H. H. Bauschke and P. L. Combettes,Convex Analysis and Monotone Operator Theory in Hilbert Spaces(CMS Books in Mathematics), 2nd. Springer, 2017

    work page 2017

  32. [32]

    G. H. Golub and C. F. Van Loan,Matrix Computations(Johns Hop- kins Studies in the Mathematical Sciences), Fourth edition. Baltimore: The Johns Hopkins University Press, 2013, 756 pp

    work page 2013

  33. [33]

    Some interlacing properties of the schur complement of a hermitian matrix

    R. L. Smith, “Some interlacing properties of the schur complement of a hermitian matrix”,Linear Algebra and its Applications, vol. 177, pp. 137–144, 1992

    work page 1992

  34. [34]

    Harnessing sparsity over the continuum: Atomic norm minimization for superresolution

    Y . Chi and M. Ferreira Da Costa, “Harnessing sparsity over the continuum: Atomic norm minimization for superresolution”,IEEE Signal Processing Magazine, vol. 37, no. 2, pp. 39–57, 2020

    work page 2020

  35. [35]

    On the stable resolution limit of total variation regularization for spike deconvolution

    M. Ferreira Da Costa and Y . Chi, “On the stable resolution limit of total variation regularization for spike deconvolution”,IEEE Transac- tions on Information Theory, vol. 66, no. 11, pp. 7237–7252, 2020

    work page 2020

  36. [36]

    Weak sparse superresolution is well- conditioned

    M. Hockmann and S. Kunis, “Weak sparse superresolution is well- conditioned”,SIAM Journal on Imaging Sciences, vol. 16, no. 1, SC1– SC13, 2023

    work page 2023

  37. [37]

    The Geometry of Off-the-Grid Compressed Sensing

    C. Poon, N. Keriven, and G. Peyré, “The Geometry of Off-the-Grid Compressed Sensing”,Foundations of Computational Mathematics, vol. 23, no. 1, pp. 241–327, Feb. 1, 2023

    work page 2023

  38. [38]

    Reconstruction de réponses impulsionnelles sur une variété pour la spectroscopie de plasma induit par laser

    S. Michelena, J. Picheral, and M. Ferreira Da Costa, “Reconstruction de réponses impulsionnelles sur une variété pour la spectroscopie de plasma induit par laser”, inProceedings of the XXXe Colloque Francophone de Traitement du Signal et des Images (GRETSI 2025), GRETSI, Strasbourg, France, Aug. 2025

    work page 2025