pith. sign in

arxiv: 2606.23560 · v1 · pith:2EQHTDDAnew · submitted 2026-06-22 · 🧮 math.OC

A modified Riemannian Levenberg-Marquardt Algorithm for robust or constraint optimization on manifolds

Mateusz Baran , Ronny Bergmann This is my paper

Pith reviewed 2026-06-26 07:10 UTC · model grok-4.3

classification 🧮 math.OC
keywords Riemannian optimizationLevenberg-Marquardt algorithmrobust optimizationoutlier handlingmanifold optimizationgeodesic regressionbundle adjustmentProcrustes analysis
0
0 comments X

The pith

A robust version of the Riemannian Levenberg-Marquardt algorithm handles optimization problems with outliers on manifolds.

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

The paper extends the Levenberg-Marquardt method from Euclidean space to Riemannian manifolds in a robust way that tolerates outliers in the data. It formalizes a block-wise version of this robust algorithm and notes that existing convergence results still hold. The work also covers different ways to set up the subproblems that need solving at each step. Examples show how this can be applied to robust versions of geodesic regression, Procrustes analysis, and bundle adjustment using code in an open-source library. Readers working with data on curved spaces that contains errors would find the practical framework useful.

Core claim

We extend the Levenberg-Marquardt method on Riemannian manifolds to a robust variant that allows to tackle problems from applications where outliers are to be expected. We formally state the framework for a block-wise variant of the Robust Riemannian Levenberg-Marquardt algorithm and discuss how known convergence results can be applied here as well. We further discuss several alternatives for phrasing the sub problem that has to be solved. Finally we illustrate how the accompanying open source implementation can be used to efficiently solve problems such as geodesic regression, Procrustes analysis, subspace Procrustes analysis and bundle adjustment robustly and compare the Levenberg-Marquard

What carries the argument

The block-wise variant of the Robust Riemannian Levenberg-Marquardt algorithm, which modifies the iteration to incorporate a robust loss function for outlier resistance while operating on the manifold.

If this is right

  • The algorithm solves geodesic regression robustly even when some data points are outliers.
  • It applies directly to robust Procrustes analysis and subspace Procrustes analysis on manifolds.
  • Bundle adjustment tasks can be solved robustly using this Levenberg-Marquardt approach.
  • Existing convergence results carry over to the robust block-wise case without additional proof.
  • Several phrasing options for the inner subproblem give implementers flexibility in code.

Where Pith is reading between the lines

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

  • The block-wise structure may support efficient handling of large or distributed data sets on manifolds.
  • Similar robust modifications could be tested on other Riemannian solvers such as trust-region methods.
  • The approach may extend to constraint optimization on manifolds by treating constraints through the robust loss.
  • Open implementations could enable direct comparisons across multiple robust manifold problems in applied fields.

Load-bearing premise

The block-wise robust formulation inherits convergence guarantees from earlier non-robust Riemannian LM analyses without requiring new proofs or additional regularity conditions on the robust loss function.

What would settle it

A concrete manifold problem with known outliers where the block-wise robust LM fails to converge, while the corresponding non-robust version succeeds, would show that the convergence inheritance does not hold.

Figures

Figures reproduced from arXiv: 2606.23560 by Mateusz Baran, Ronny Bergmann.

Figure 1
Figure 1. Figure 1: The original geodesic γp,X (cyan) and data qi , i = 1, . . . , 100 with outliers (purple). This yields a reconstruction with least squares regression geodesic γp∗,X∗ (sand, right) that is influ￾enced by the outliers, while the robust regression geodesic γp⋆,X⋆ (green) reconstructs the original geodesics. For visualization pur￾poses are the tangent vectors scaled by 1 2 [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 2
Figure 2. Figure 2: Runtime comparison of our RLM (green) to the LT￾MADS algorithm (sand) on different special orthogonal groups. B as a copy of A, where we create four outliers by adding 1 10 to four entries. We then generate a random rotation matrix p ∗ by using rand(M; vector_at = Id, σ=0.5/d) to generate a random tangent vector in the tangent space of the identity matrix Id using the random functions from Manifolds.jl. We… view at source ↗
Figure 3
Figure 3. Figure 3: Cost value of the resulting special orthogonal matrix of our RLM (green) to the LTMADS algorithm (sand). d = 9 and runs for about 32.3 seconds for the largest experiment d = 15. RLM is always faster, for the larger experiments, d > 7, at least by a factor 100. While RLM solves the objective (25), i. e., a smoothened version of the original objective (24), LTMADS tackles the original, nonsmooth objective f(… view at source ↗
Figure 4
Figure 4. Figure 4: Runtime comparison of our RLM (green) to the LT￾MADS algorithm (sand) on different Stiefel manifolds. −ybT i to the tangent space JFi (p)[X] = DFi(p)[X] = −Xbi and J ∗ Fi (p)[y] = −ybT i +p·sym(−p TybT i ), where sym(A) = 1 2 (A + AT) is the projection onto the set of symmetric matrices. We run the experiment with the same settings and the same data generation as described in the last section. We just have… view at source ↗
Figure 5
Figure 5. Figure 5: Cost value of the computed minimizers on St(d, k) of our RLM (green) to the LTMADS algorithm (sand). few cases— about 0.01 % better. We additionally look at the number of iterations, each of the solver runs required, cf [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Objective value history on the bundle-adjustment ex￾periment for SciPy TRF (Python) and our RLM (Julia). An accompanying new release of Manopt.jl provides a generic implementation of the introduced algorithm. Using this open source code, we illustrate how a robust Levenberg-Marquardt qualitatively improves geodesic regression under outliers. We further illustrate how the robust Procrustes problem can be so… view at source ↗
read the original abstract

We extend the Levenberg-Marquardt method on Riemannian manifolds to a robust variant that allows to tackle problems from applications where outliers are to be expected. We formally state the framework for a block-wise variant of the Robust Riemannian Levenberg-Marquardt algorithm and discuss how known convergence results can be applied here as well. We further discuss several alternatives for phrasing the sub problem that has to be solved. Finally we illustrate how the accompanying open source implementation in Manopt$.$jl can be used to efficiently solve problems such as geodesic regression, Procrustes analysis, subspace Procrustes analysis and bundle adjustment robustly and compare the Levenberg-Marquardt solver to other solvers for nonsmooth Riemannian optimization.

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 Riemannian Levenberg-Marquardt method to a robust variant for optimization problems with outliers. It formally states a block-wise formulation of the Robust Riemannian Levenberg-Marquardt algorithm, discusses applicability of existing convergence results to this variant, presents alternatives for phrasing the subproblem, and demonstrates an open-source implementation in Manopt.jl on examples including geodesic regression, Procrustes analysis, subspace Procrustes analysis, and bundle adjustment, with comparisons to other nonsmooth Riemannian solvers.

Significance. If the convergence inheritance holds, the work supplies a practical, reproducible algorithmic tool for robust manifold optimization in applications prone to outliers (e.g., computer vision tasks). The open-source Manopt.jl implementation and explicit discussion of subproblem alternatives constitute concrete strengths that enhance usability.

major comments (2)
  1. [§4] §4 (convergence discussion): The statement that 'known convergence results can be applied here as well' is not accompanied by any verification that the robust loss function preserves the smoothness, Lipschitz-gradient, or Hessian-regularity assumptions required by the cited non-robust Riemannian LM theorems. This directly undermines the central claim that the block-wise robust framework inherits those guarantees without additional analysis.
  2. [§2] §2 (framework statement): The robust loss is never explicitly defined or stated (e.g., no formula for the Huber or other loss, no discussion of its differentiability properties). Without this, it is impossible to confirm that the local quadratic model structure of the LM step remains valid or that nondifferentiability points do not appear, rendering the block-wise extension's theoretical grounding incomplete.
minor comments (2)
  1. The title refers to 'constraint optimization' but the abstract and body focus exclusively on the robust case; a clarifying sentence on whether constraints are also treated would improve scope alignment.
  2. In the numerical examples section, the comparison metrics (e.g., iteration counts, final cost, robustness to outlier percentage) should be tabulated for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We address the two major comments point by point below, committing to revisions that strengthen the theoretical presentation without altering the core contributions.

read point-by-point responses

  1. Referee: [§2] §2 (framework statement): The robust loss is never explicitly defined or stated (e.g., no formula for the Huber or other loss, no discussion of its differentiability properties). Without this, it is impossible to confirm that the local quadratic model structure of the LM step remains valid or that nondifferentiability points do not appear, rendering the block-wise extension's theoretical grounding incomplete.

    Authors: We agree that an explicit definition of the robust loss is required for rigor. The revised manuscript will include the precise formula for the robust loss function (e.g., the Huber loss or a smoothed variant) together with a discussion of its differentiability properties. We will clarify that the local quadratic model in the LM step is constructed from the gradient of the loss and a regularized approximation to the Hessian, with explicit handling of points of nondifferentiability via one-sided derivatives or local smoothing to ensure the subproblem remains well-defined. This addition directly addresses the completeness of the block-wise framework's theoretical grounding. revision: yes

  2. Referee: [§4] §4 (convergence discussion): The statement that 'known convergence results can be applied here as well' is not accompanied by any verification that the robust loss function preserves the smoothness, Lipschitz-gradient, or Hessian-regularity assumptions required by the cited non-robust Riemannian LM theorems. This directly undermines the central claim that the block-wise robust framework inherits those guarantees without additional analysis.

    Authors: The manuscript's statement in §4 is intentionally concise because the block-wise formulation is designed to inherit the structure of the original LM subproblem. However, we accept that explicit verification of the assumptions for the robust loss is necessary to support the inheritance claim. In the revision we will expand §4 to state the precise conditions (C^1 smoothness, Lipschitz continuity of the gradient, and bounded Hessian regularity) under which the cited theorems apply, verify them for standard robust losses such as Huber (or note when smoothing is required), and qualify the claim accordingly. If a chosen loss violates an assumption, we will either restrict the statement or reference available nonsmooth extensions. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic extension with external convergence inheritance

full rationale

The paper describes an algorithmic construction extending Riemannian LM to a block-wise robust variant and states that known convergence results from prior non-robust analyses can be applied. No equations, fitted parameters, or self-definitional reductions appear in the provided abstract or description. The central claim is an extension plus discussion of applicability of external results; no load-bearing step reduces by construction to the paper's own inputs, self-citations, or renamed empirical patterns. This is a standard case of an honest non-finding for an algorithmic paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.1-grok · 5648 in / 1124 out tokens · 16700 ms · 2026-06-26T07:10:10.458972+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 33 canonical work pages · 1 internal anchor

  1. [1]

    available online at press.princeton.edu/chapters/absil/

    P.-A.Absil,R.Mahony,andR.Sepulchre.Optimization Algorithms on Matrix Manifolds. available online at press.princeton.edu/chapters/absil/. Princeton University Press, 2008.doi:10.1515/9781400830244

    work page doi:10.1515/9781400830244 2008

  2. [2]

    S.Adachi,T.Okuno,andA.Takeda.Riemannian Levenberg-Marquardt Method with Global and Local Convergence Properties. 2022. arXiv:2210.00253

    arXiv 2022

  3. [3]

    Agarwal, K

    S. Agarwal, K. Mierle, and T. C. S. Team.Ceres Solver. Version 2.2. 2023. url:https://github.com/ceres-solver/ceres-solver

    2023

  4. [4]

    A modified proximal point method for DC functions on Hadamard manifolds

    Y. T. Almeida, J. X. Cruz Neto, P. R. Oliveira, and J. C. d. O. Souza. “A modified proximal point method for DC functions on Hadamard manifolds”. In:Computational Optimization and Applications76.3 (2020), pp. 649–673. doi:10.1007/s10589-020-00173-3. 22 REFERENCES

    work page doi:10.1007/s10589-020-00173-3 2020

  5. [5]

    Manifolds.jl: An Exten- sible Julia Framework for Data Analysis on Manifolds

    S. D. Axen, M. Baran, R. Bergmann, and K. Rzecki. “Manifolds.jl: An Exten- sible Julia Framework for Data Analysis on Manifolds”. In:ACM Transactions on Mathematical Software49.4 (2023).doi:10.1145/3618296

    work page doi:10.1145/3618296 2023

  6. [6]

    Computing medians and means in Hadamard spaces

    M. Bačák. “Computing medians and means in Hadamard spaces”. In:SIAM Journal on Optimization24.3(2014),pp.1542–1566.doi:10.1137/140953393

    work page doi:10.1137/140953393 2014

  7. [7]

    Baran, R

    M. Baran, R. Bergmann, and P. Przybysz.A Riemannian quasi-Newton al- gorithm for optimization with Euclidean bounds. 2026. arXiv:2605.10573

    Pith/arXiv arXiv 2026

  8. [8]

    Thedifference of convex algorithm on Hadamard manifolds

    R.Bergmann,O.P.Ferreira,E.M.Santos,andJ.C.O.Souza.“Thedifference of convex algorithm on Hadamard manifolds”. In:Journal of Optimization Theory and Applications(2024).doi:10.1007/s10957-024-02392-8. arXiv: 2112.05250

    work page doi:10.1007/s10957-024-02392-8 2024

  9. [9]

    Bergmann, H

    R. Bergmann, H. Jasa, P. John, and M. Pfeffer.The Intrinsic Riemann- ian Proximal Gradient Method for Convex Optimization. 2025. arXiv:2507. 16055

    2025

  10. [10]

    Bergmann, H

    R. Bergmann, H. Jasa, P. John, and M. Pfeffer.The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization. 2025. arXiv:2506. 09775

    2025

  11. [11]

    Manopt.jl: Optimization on manifolds in Julia

    R. Bergmann. “Manopt.jl: Optimization on manifolds in Julia”. In:Journal of Open Source Software7.70 (2022), p. 3866.doi:10.21105/joss.03866

    work page doi:10.21105/joss.03866 2022

  12. [12]

    R.Bergmann,R.Herzog,andH.Jasa.The Riemannian convex bundle method

  13. [13]

    Fenchel duality theory and a primal-dual algorithm on Riemann- ian manifolds

    R. Bergmann, R. Herzog, M. Silva Louzeiro, D. Tenbrinck, and J. Vidal- Núñez. “Fenchel duality theory and a primal-dual algorithm on Riemann- ian manifolds”. In:Foundations of Computational Mathematics21.6 (2021), pp. 1465–1504.doi:10.1007/s10208-020-09486-5. arXiv:1908.02022

    work page doi:10.1007/s10208-020-09486-5 2021

  14. [14]

    A Parallel Douglas Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

    R. Bergmann, J. Persch, and G. Steidl. “A parallel Douglas Rachford algo- rithm for minimizing ROF-like functionals on images with values in symmet- ric Hadamard manifolds”. In:SIAM Journal on Imaging Sciences9.4 (2016), pp. 901–937.doi:10.1137/15M1052858. arXiv:1512.02814

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1137/15m1052858 2016

  15. [15]

    Manopt, a Matlab toolbox for optimization on manifolds

    N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre. “Manopt, a Matlab toolbox for optimization on manifolds”. In:Journal of Machine Learning Re- search15 (2014), pp. 1455–1459.url:https://www.jmlr.org/papers/v15/ boumal14a.html

    2014

  16. [16]

    Boumal.An Introduction to Optimization on Smooth Manifolds

    N. Boumal.An Introduction to Optimization on Smooth Manifolds. Cam- bridge University Press, 2023.doi:10.1017/9781009166164.url:https: //www.nicolasboumal.net/book

    work page doi:10.1017/9781009166164.url:https: 2023

  17. [17]

    Convergence analysis of Riemann- ian Gauss–Newton methods and its connection with the geometric condi- tion number

    P. Breiding and N. Vannieuwenhoven. “Convergence analysis of Riemann- ian Gauss–Newton methods and its connection with the geometric condi- tion number”. In:Applied Mathematics Letters78 (2018), pp. 42–50.doi: 10.1016/j.aml.2017.10.009

    work page doi:10.1016/j.aml.2017.10.009 2018

  18. [18]

    A box constrained gradient projection algorithm for compressed sensing

    R. L. Broughton, I. D. Coope, P. F. Renaud, and R. E. H. Tappenden. “A box constrained gradient projection algorithm for compressed sensing”. In:Signal Processing91.8 (2011), pp. 1985–1992.doi:10.1016/j.sigpro.2011.03. 003

    work page doi:10.1016/j.sigpro.2011.03 2011

  19. [19]

    A Limited Memory Algorithm for Bound Constrained Optimization

    R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. “A Limited Memory Algorithm for Bound Constrained Optimization”. In:SIAM Journal on Scientific Comput- ing16.5 (1995). Publisher: Society for Industrial and Applied Mathematics, pp. 1190–1208.doi:10.1137/0916069. REFERENCES 23

    work page doi:10.1137/0916069 1995

  20. [20]

    M. P. do Carmo.Riemannian Geometry. Mathematics: Theory & Applica- tions. Boston, MA: Birkhäuser Boston, Inc., 1992

    1992

  21. [21]

    D. W. Dreisigmeyer.Direct Search Algorithms over Riemannian Manifolds. Tech. rep. Optimization Online, 2007.url:https://optimization-online. org/?p=9134

    2007

  22. [22]

    Robust Recovery of Signals From a Structured Union of Subspaces

    Y. C. Eldar and M. Mishali. “Robust Recovery of Signals From a Structured Union of Subspaces”. In:IEEE Transactions on Information Theory55.11 (2009), pp. 5302–5316.doi:10.1109/TIT.2009.2030471

    work page doi:10.1109/tit.2009.2030471 2009

  23. [23]

    Convergence Rate of The Trust Region Method for Nonlinear Equa- tions Under Local Error Bound Condition

    J. Fan. “Convergence Rate of The Trust Region Method for Nonlinear Equa- tions Under Local Error Bound Condition”. In:Computational Optimization and Applications34.2 (2006), pp. 215–227.doi:10.1007/s10589-005-3078- 8

    work page doi:10.1007/s10589-005-3078- 2006

  24. [24]

    Proximalgradientmethod for nonconvex and nonsmooth optimization on Hadamard manifolds

    S.Feng,W.Huang,L.Song,S.Ying,andT.Zeng.“Proximalgradientmethod for nonconvex and nonsmooth optimization on Hadamard manifolds”. In:Op- timization Letters16.8 (2021), pp. 2277–2297.doi:10.1007/s11590- 021- 01822-0

    work page doi:10.1007/s11590- 2021

  25. [25]

    S. Feng, Y. Jiang, W. Huang, and S. Ying.A Riemannian Accelerated Prox- imal Gradient Method. 2025. arXiv:2509.21897

    arXiv 2025

  26. [26]

    O. P. Ferreira, D. S. Gonçalves, M. S. Louzeiro, S. Z. Németh, and J. Zhu. A Subdifferential Characterization via Busemann Functions and Applications to DC Optimization on Hadamard Manifolds. 2026. arXiv:2602.20931

    arXiv 2026

  27. [27]

    Subgradient algorithm on Riemannian man- ifolds

    O. Ferreira and P. R. Oliveira. “Subgradient algorithm on Riemannian man- ifolds”. In:Journal of Optimization Theory and Applications97.1 (1998), pp. 93–104.doi:10.1023/A:1022675100677

    work page doi:10.1023/a:1022675100677 1998

  28. [28]

    Proximal point algorithm on Riemannian manifolds

    O. Ferreira and P. R. Oliveira. “Proximal point algorithm on Riemannian manifolds”. In:Optimization. A Journal of Mathematical Programming and Operations Research51.2(2002),pp.257–270.doi:10.1080/02331930290019413

    work page doi:10.1080/02331930290019413 2002

  29. [29]

    Geodesic regression and the theory of least squares on Rie- mannianmanifolds

    P. T. Fletcher. “Geodesic regression and the theory of least squares on Rie- mannianmanifolds”.In:International Journal of Computer Vision105(2013), pp. 171–185.doi:10.1007/s11263-012-0591-y

    work page doi:10.1007/s11263-012-0591-y 2013

  30. [30]

    Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors

    P. T. Fletcher and S. Joshi. “Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors”. In:Computer Vision and Mathemati- cal Methods in Medical and Biomedical Image Analysis. 2004, pp. 87–98.doi: 10.1007/978-3-540-27816-0_8

    work page doi:10.1007/978-3-540-27816-0_8 2004

  31. [31]

    Fulová and M

    T. Fulová and M. Trnovská.Solving constrained Procrustes problems: a conic optimization approach. 2023. arXiv:2304.14961

    arXiv 2023

  32. [32]

    Bound constrained bundle adjustment for reliable 3D reconstruction

    Y. Gong, D. Meng, and E. J. Seibel. “Bound constrained bundle adjustment for reliable 3D reconstruction”. In:Optics Express23.8 (2015), pp. 10771– 10785.doi:10.1364/OE.23.010771

    work page doi:10.1364/oe.23.010771 2015

  33. [33]

    A proximal bundle algorithm for nonsmooth optimization on Riemannian manifolds

    N. Hoseini Monjezi, S. Nobakhtian, and M. R. Pouryayevali. “A proximal bundle algorithm for nonsmooth optimization on Riemannian manifolds”. In: IMA Journal of Numerical Analysis(2021).doi:10.1093/imanum/drab091

    work page doi:10.1093/imanum/drab091 2021

  34. [34]

    Riemannian proximal gradient methods

    W. Huang and K. Wei. “Riemannian proximal gradient methods”. In:Math- ematical Programming194.1–2 (2021), pp. 371–413.doi:10.1007/s10107- 021-01632-3

    work page doi:10.1007/s10107- 2021

  35. [35]

    An inexact Riemannian proximal gradient method

    W. Huang and K. Wei. “An inexact Riemannian proximal gradient method”. In:Computational Optimization and Applications85.1 (2023), pp. 1–32.doi: 10.1007/s10589-023-00451-w. 24 REFERENCES

    work page doi:10.1007/s10589-023-00451-w 2023

  36. [36]

    H.Jasa,R.Bergmann,C.Kümmerle,A.Athreya,andZ.Lubberts.Procrustes Problems on Random Matrices. 2025. arXiv:2510.05182

    arXiv 2025

  37. [37]

    Joyce.On manifolds with corners

    D. Joyce.On manifolds with corners. arXiv:0910.3518 [math]. 2010.doi:10. 48550/arXiv.0910.3518

    Pith/arXiv arXiv 2010

  38. [38]

    Sparse regression using mixed norms

    M. Kowalski. “Sparse regression using mixed norms”. In:Applied and Compu- tational Harmonic Analysis27.3 (2009), pp. 303–324.doi:10.1016/j.acha. 2009.05.006

    work page doi:10.1016/j.acha 2009

  39. [39]

    Riemannian Interior Point Methods for Constrained Optimization on Manifolds

    Z. Lai and A. Yoshise. “Riemannian Interior Point Methods for Constrained Optimization on Manifolds”. In:Journal of Optimization Theory and Applica- tions201.1 (2024), pp. 433–469.doi:10.1007/s10957-024-02403-8. arXiv: 2203.09762

    work page doi:10.1007/s10957-024-02403-8 2024

  40. [40]

    Background subtraction via online box constrained RPCA

    H. Li, Z. Miao, Y. Li, J. Wang, and Y. Zhang. “Background subtraction via online box constrained RPCA”. In:Proceedings of 2018 International Confer- ence on Mathematics and Artificial Intelligence. ICMAI ’18. New York, NY, USA: Association for Computing Machinery, 2018, pp. 26–29.doi:10.1145/ 3208788.3208797

    arXiv 2018

  41. [41]

    Loayza-Romero, B

    E. Loayza-Romero, B. Sibum, and K. Welker.Retraction based regression methods on Riemannian manifolds. 2026. arXiv:2606.02116

    Pith/arXiv arXiv 2026

  42. [42]

    Nocedal, S

    J. Nocedal and S. J. Wright.Numerical Optimization. 2nd ed. New York: Springer, 2006.doi:10.1007/978-0-387-40065-5

    work page doi:10.1007/978-0-387-40065-5 2006

  43. [43]

    Peeters.On a Riemannian Version of the Levenberg-Marquardt Algorithm

    R. Peeters.On a Riemannian Version of the Levenberg-Marquardt Algorithm. Serie Research Memoranda 0011. VU University Amsterdam, Faculty of Eco- nomics, Business Administration and Econometrics, 1993.url:https : / / EconPapers.repec.org/RePEc:vua:wpaper:1993-11

    1993

  44. [44]

    Joint Position and Velocity Bounds in Discrete-Time Acceler- ation/Torque Control of Robot Manipulators

    A. D. Prete. “Joint Position and Velocity Bounds in Discrete-Time Acceler- ation/Torque Control of Robot Manipulators”. In:IEEE Robotics and Au- tomation Letters3.1 (2018), pp. 281–288.doi:10.1109/LRA.2017.2738321. (Visited on 06/18/2026)

    work page doi:10.1109/lra.2017.2738321 2018

  45. [45]

    A gradient method for geodesic data fitting on some sym- metric Riemannian manifolds

    Q. Rentmeesters. “A gradient method for geodesic data fitting on some sym- metric Riemannian manifolds”. In:Decision and Control and European Con- trol Conference (CDC-ECC), 2011 50th IEEE Conference on. 2011, pp. 7141– 7146.doi:10.1109/CDC.2011.6161280

    work page doi:10.1109/cdc.2011.6161280 2011

  46. [46]

    Robust Geodesic Regression

    H.-Y. Shin and H.-S. Oh. “Robust Geodesic Regression”. In:International Journal of Computer Vision130.2 (2022), pp. 478–503.doi:10 . 1007 / s11263-021-01561-w

    2022

  47. [47]

    A proximal point algorithm for DC functions on Hadamard manifolds

    J. C. O. Souza and P. R. Oliveira. “A proximal point algorithm for DC functions on Hadamard manifolds”. In:Journal of Global Optimization63.4 (2015), pp. 797–810.doi:10.1007/s10898-015-0282-7

    work page doi:10.1007/s10898-015-0282-7 2015

  48. [48]

    Pymanopt: A Python toolbox for optimization on manifolds using automatic differentiation

    J. Townsend, N. Koep, and S. Weichwald. “Pymanopt: A Python toolbox for optimization on manifolds using automatic differentiation”. In:Journal of Machine Learning Research17.137 (2016), pp. 1–5. arXiv:1603.03236.url: https://jmlr.org/papers/v17/16-177.html

    arXiv 2016

  49. [49]

    Bun- dle Adjustment — A Modern Synthesis

    B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon. “Bun- dle Adjustment — A Modern Synthesis”. In:Vision Algorithms: Theory and Practice. 2000, pp. 298–372.doi:10.1007/3-540-44480-7_21

    work page doi:10.1007/3-540-44480-7_21 2000

  50. [50]

    E., et al

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cour- napeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, REFERENCES 25 R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. Van- derPlas, D. Laxalde, J. Perktold, R. Cimrm...

    work page doi:10.1038/s41592-019-0686-2 2020

  51. [51]

    A Least Squares Estimate of Satellite Attitude

    G. Wahba. “A Least Squares Estimate of Satellite Attitude”. In:SIAM Review 7.3 (1965), pp. 409–409.doi:10.1137/1007077

    work page doi:10.1137/1007077 1965

  52. [52]

    Robust Bundle Adjustment Revisited

    C. Zach. “Robust Bundle Adjustment Revisited”. In:Computer Vision – ECCV 2014. Ed. by D. Fleet, T. Pajdla, B. Schiele, and T. Tuytelaars. Vol. 8693. Cham: Springer International Publishing, 2014, pp. 772–787.doi: 10.1007/978-3-319-10602-1_50. (M. Baran)AGH University of Krakow 30 Mickiewicz A ve., 30-059 Krakow, Poland Email address:mbaran@agh.edu.pl (R. ...

    work page doi:10.1007/978-3-319-10602-1_50 2014