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arxiv: 2605.23097 · v1 · pith:FCJNFJYHnew · submitted 2026-05-21 · 🧮 math.OC · math.GT

Proximal DCA for Fr\'echet Regression on Riemannian Manifolds with Bounded Curvature

Yamin Zhou , C\'esar A. Uribe This is my paper

Pith reviewed 2026-05-25 05:09 UTC · model grok-4.3

classification 🧮 math.OC math.GT
keywords Fréchet regressionRiemannian manifoldproximal DC algorithmsigned barycenterbounded curvatureconvergence analysisoptimization on manifolds
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The pith

Signed Fréchet regression on Riemannian manifolds with bounded curvature reduces to a proximal DC problem solved by the FRIDA algorithm with convergence guarantees.

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

The paper shows how to compute Fréchet regression fits when responses lie on a Riemannian manifold and the regression weights may be positive or negative. It restricts attention to a strongly convex normal ball around the data and uses local smoothness plus curvature comparison to rewrite the objective as a difference-of-convex program. The resulting FRIDA algorithm is proved to produce iterates that descend and converge to stationary points, with explicit dependence on the curvature bounds and signed-weight conditions.

Core claim

The authors formulate signed Fréchet regression as a locally controlled Riemannian proximal DC problem by restricting optimization to a strongly convex normal ball containing the response support and applying Hessian comparison together with Jacobi-field estimates. This yields the FRIDA algorithm (exact and inexact versions) for which they establish existence and interiority of minimizers under explicit signed-weight conditions, curvature-dependent strong convexity of the proximal subproblems, descent and convergence of the iterates to stationary points, sublinear complexity estimates, and full-sequence convergence with KL-type local rates under real-analyticity.

What carries the argument

The Riemannian proximal DC problem obtained by restricting to a strongly convex normal ball and applying local smoothness, Hessian comparison, and Jacobi-field estimates.

If this is right

  • Minimizers exist and lie in the interior of the normal ball under explicit signed-weight conditions.
  • Each proximal subproblem is strongly convex with a modulus that depends on the sectional-curvature bounds.
  • The exact and inexact FRIDA iterates exhibit descent and converge to stationary points.
  • Sublinear complexity bounds hold for the algorithm.
  • Under real-analyticity the whole sequence converges with KL-type local rates.

Where Pith is reading between the lines

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

  • The local-ball restriction may let similar DC reformulations be written for other nonconvex manifold-valued estimation problems whose objectives become difference-of-convex only locally.
  • The curvature-dependent strong-convexity modulus supplies a concrete way to compare iteration counts against Euclidean DC methods when the manifold is replaced by the sphere or hyperbolic space.
  • The convergence theory suggests checking whether the same signed-weight conditions suffice for related tasks such as manifold-valued quantile regression.

Load-bearing premise

The response support lies inside a strongly convex normal ball so that local estimates control the objective globally.

What would settle it

Numerical runs on the two-sphere with a signed-weight vector satisfying the paper's interiority condition in which the FRIDA iterates fail to approach any stationary point.

Figures

Figures reproduced from arXiv: 2605.23097 by C\'esar A. Uribe, Yamin Zhou.

Figure 1
Figure 1. Figure 1: Illustration of Fréchet regression on four manifolds: the sphere, torus, cylinder, and elliptic [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Global Fréchet Regression on data lying on a geodesic of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (c)–(d) shows the spiral-noise experiment on S 2 . Panel (c) gives the longitude–latitude projec￾tion, and Panel (d) maps the same curves onto the sphere. The true spiral response is shown in blue, the local Fréchet estimate in orange, and the global Fréchet estimate in purple. Noisy observations are plotted in green, and the initialization is marked by a star. Compared with the global estimator, the local… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between Riemannian gradient descent and FRIDA on [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fréchet regression experiments on the embedded torus. Panels (a)–(b) show global Fréchet regres [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

Fr\'echet regression generalizes linear regression to metric-space-valued responses by defining fitted values as minimizers of weighted Fr\'echet functionals. Since these weights may have mixed signs, the resulting objective is a signed barycenter problem rather than a standard convex barycenter problem. On Riemannian manifolds, this is further complicated by the lack of global geodesic convexity and possible nonsmoothness of squared distances near cut loci. We study signed Fr\'echet regression on complete manifolds with two-sided bounded sectional curvature. By restricting optimization to a strongly convex normal ball containing the response support, we use local smoothness, Hessian comparison, and Jacobi-field estimates to formulate the problem as a locally controlled Riemannian proximal DC problem. This leads to FRIDA (Fr\'echet Regression via Riemannian Iterative DC Algorithm), an exact and inexact proximal DC algorithm for computing regression fits. We prove existence and interiority of minimizers under explicit signed-weight conditions, establish curvature-dependent strong convexity of the proximal subproblems, and show descent and convergence of the iterates to stationary points. We also derive sublinear complexity estimates and, under real-analyticity, obtain full-sequence convergence with KL-type local rates. These results provide a rigorous optimization framework for signed Fr\'echet regression on manifolds with bounded curvature.

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 proposes FRIDA, a proximal difference-of-convex (DC) algorithm for computing signed Fréchet regression on Riemannian manifolds with bounded sectional curvature. By localizing the problem to a strongly convex normal ball containing the response support and using Hessian comparison and Jacobi field estimates, the authors formulate the regression as a Riemannian proximal DC problem. They prove existence and interiority of minimizers under explicit signed-weight conditions, curvature-dependent strong convexity of the proximal subproblems, descent properties, convergence to stationary points, sublinear complexity bounds, and KL-type rates under real-analyticity assumptions.

Significance. If the localization to the normal ball is preserved by the algorithm iterates, the results would provide a rigorous and practical optimization method for non-convex Fréchet regression problems on manifolds, extending beyond standard convex barycenter settings to handle mixed-sign weights and manifold geometry.

major comments (2)
  1. [Abstract and algorithm description] Abstract and § on the Riemannian proximal DC formulation: the central construction restricts the entire problem to a strongly convex normal ball B containing the response support and invokes local smoothness, Hessian comparison theorems, and Jacobi-field estimates to obtain curvature-dependent strong convexity. Nothing establishes that the exact or inexact proximal updates (or the DC linearization steps) map B into itself; if an iterate leaves B the curvature bounds cease to control the Hessian of the squared-distance terms and the strong-convexity constants become invalid. This invariance is load-bearing for the existence/interiority, descent, and convergence results.
  2. [Convergence and complexity sections] Convergence and complexity sections: the descent property, convergence to stationary points, and sublinear complexity estimates all rely on the proximal subproblems remaining strongly convex with uniform curvature-dependent constants; without a proof that iterates remain inside B these claims require additional justification or a global extension of the estimates.
minor comments (2)
  1. The explicit signed-weight conditions for interiority of minimizers would benefit from a short numerical example illustrating the boundary case.
  2. Notation for the proximal mapping and the DC decomposition could be made more uniform across the exact and inexact variants.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and for highlighting the critical role of ball invariance. We agree this property is essential for the curvature-dependent estimates and will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and algorithm description] Abstract and § on the Riemannian proximal DC formulation: the central construction restricts the entire problem to a strongly convex normal ball B containing the response support and invokes local smoothness, Hessian comparison theorems, and Jacobi-field estimates to obtain curvature-dependent strong convexity. Nothing establishes that the exact or inexact proximal updates (or the DC linearization steps) map B into itself; if an iterate leaves B the curvature bounds cease to control the Hessian of the squared-distance terms and the strong-convexity constants become invalid. This invariance is load-bearing for the existence/interiority, descent, and convergence results.

    Authors: We acknowledge that the current manuscript does not contain an explicit invariance proof for the proximal and DC-linearization steps. This is a substantive gap. In the revision we will add a dedicated lemma establishing that, under the signed-weight conditions already used for interiority of the minimizer, both the exact and inexact proximal operators map B into itself. The argument will combine the strong convexity of the proximal subproblem (already proved via Hessian comparison) with a contraction estimate on the proximal mapping inside B, together with a uniform bound on the linearization error that prevents escape. We will also insert a short paragraph in the algorithm section clarifying that all iterates are confined to B by construction once this lemma is in place. revision: yes

  2. Referee: [Convergence and complexity sections] Convergence and complexity sections: the descent property, convergence to stationary points, and sublinear complexity estimates all rely on the proximal subproblems remaining strongly convex with uniform curvature-dependent constants; without a proof that iterates remain inside B these claims require additional justification or a global extension of the estimates.

    Authors: We agree that the descent, stationarity, and complexity results are conditional on the iterates remaining inside B. Once the invariance lemma described above is added, the existing curvature-dependent strong-convexity constants remain valid throughout the iteration, and the descent and convergence arguments carry through unchanged. We will explicitly reference the new lemma in the statements of the descent and complexity theorems. If the referee prefers, we can also include a brief remark on how the same estimates could be globalized via the two-sided curvature bound, but we believe the localized approach with invariance is the most direct and practical route. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external Riemannian geometry estimates

full rationale

The paper formulates the signed Fréchet regression problem by restricting to a strongly convex normal ball and invoking local smoothness, Hessian comparison theorems, and Jacobi-field estimates to obtain curvature-dependent strong convexity for the proximal subproblems. These steps rely on standard comparison geometry results rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The existence, descent, and convergence claims follow from the DC algorithm framework under those external bounds, without the target results reducing to the inputs by construction. The approach is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are limited to statements explicitly invoked in the abstract.

axioms (2)
  • domain assumption Manifolds are complete with two-sided bounded sectional curvature
    Invoked to obtain local smoothness, Hessian comparison, and Jacobi-field estimates inside the normal ball.
  • domain assumption Optimization can be restricted to a strongly convex normal ball containing the response support
    Central modeling choice that converts the global problem into a locally controlled DC program.

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    On the convergence of gradient descent for finding the Riemannian center of mass.SIAM Journal on Control and Optimization, 51(3):2230–2260, 2013

    Bijan Afsari, Roberto Tron, and René Vidal. On the convergence of gradient descent for finding the Riemannian center of mass.SIAM Journal on Control and Optimization, 51(3):2230–2260, 2013

    work page 2013

  2. [2]

    Altschuler, Sinho Chewi, Patrik Gerber, and Austin J

    Jason M. Altschuler, Sinho Chewi, Patrik Gerber, and Austin J. Stromme. Averaging on the Bures– Wasserstein manifold: Dimension-free convergence of gradient descent. InAdvances in Neural Infor- mation Processing Systems, volume 34, pages 22132–22145, 2021. 24

    work page 2021

  3. [3]

    On the convergence of the proximal algorithm for nonsmooth func- tions involving analytic features.Mathematical Programming, 116(1):5–16, 2009

    Hedy Attouch and Jérôme Bolte. On the convergence of the proximal algorithm for nonsmooth func- tions involving analytic features.Mathematical Programming, 116(1):5–16, 2009

    work page 2009

  4. [4]

    The variational method of moments.Journal of the Royal Statis- tical Society Series B: Statistical Methodology, 85(3):810–841, 2023

    Andrew Bennett and Nathan Kallus. The variational method of moments.Journal of the Royal Statis- tical Society Series B: Statistical Methodology, 85(3):810–841, 2023

    work page 2023

  5. [5]

    The differ- ence of convex algorithm on Hadamard manifolds.Journal of Optimization Theory and Applications, 201(1):221–251, 2024

    Ronny Bergmann, Orizon P Ferreira, Elianderson M Santos, and João Carlos O Souza. The differ- ence of convex algorithm on Hadamard manifolds.Journal of Optimization Theory and Applications, 201(1):221–251, 2024

    work page 2024

  6. [6]

    Carmo.Riemannian geometry / Manfredo do Carmo ; translated by Francis Flaherty.Mathematics

    Manfredo Perdigao do. Carmo.Riemannian geometry / Manfredo do Carmo ; translated by Francis Flaherty.Mathematics. Theory and applications. Birkhäuser, Boston, 1992

    work page 1992

  7. [7]

    Fan and I

    J. Fan and I. Gijbels.Local polynomial modelling and its applications, volume 66 ofMonographs on Statistics and Applied Probability. Chapman & Hall, London, 1996

    work page 1996

  8. [8]

    A Bregman divergence view on the difference-of- convex algorithm

    Oisin Faust, Hamza Fawzi, and James Saunderson. A Bregman divergence view on the difference-of- convex algorithm. InInternational Conference on Artificial Intelligence and Statistics, pages 3427–

    work page

  9. [9]

    Large sample properties of generalized method of moments estimators.Economet- rica: Journal of the econometric society, pages 1029–1054, 1982

    Lars Peter Hansen. Large sample properties of generalized method of moments estimators.Economet- rica: Journal of the econometric society, pages 1029–1054, 1982

    work page 1982

  10. [10]

    Generalized differentiability/duality and optimization for problems dealing with differences of convex functions

    J-B Hiriart-Urruty. Generalized differentiability/duality and optimization for problems dealing with differences of convex functions. InConvexity and Duality in Optimization: Proceedings of the Sympo- sium on Convexity and Duality in Optimization Held at the University of Groningen, The Netherlands June 22, 1984, pages 37–70. Springer, 1985

    work page 1984

  11. [11]

    DC programming: overview.Journal of Optimization Theory and Applications, 103:1–43, 1999

    Reiner Horst and Nguyen V Thoai. DC programming: overview.Journal of Optimization Theory and Applications, 103:1–43, 1999

    work page 1999

  12. [12]

    Local Fréchet regression with spherical predictors

    Chang Jun Im, Jeong Min Jeon, and Byeong U Park. Local Fréchet regression with spherical predictors. Electronic Journal of Statistics, 19(2):5313–5367, 2025

    work page 2025

  13. [13]

    Robert T. Jantzen. Geodesics on the torus and other surfaces of revolution clarified using undergraduate physics tricks with bonus: Nonrelativistic and relativistic kepler problems, 2010. Lecture notes

    work page 2010

  14. [14]

    Proof of the gradient conjecture of R

    Krzysztof Kurdyka, Tadeusz Mostowski, and Adam Parusi ´nski. Proof of the gradient conjecture of R. Thom.Annals of Mathematics, pages 763–792, 2000

    work page 2000

  15. [15]

    DC programming and DCA: thirty years of developments

    Hoai An Le Thi and Tao Pham Dinh. DC programming and DCA: thirty years of developments. Mathematical Programming, 169(1):5–68, 2018

    work page 2018

  16. [16]

    Lee.Introduction to Smooth Manifolds

    J. Lee.Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Springer New York, 2012

    work page 2012

  17. [17]

    Springer, 2018

    John M Lee.Introduction to Riemannian manifolds, volume 2. Springer, 2018

    work page 2018

  18. [18]

    A type of nonlinear Fréchet regressions.arXiv preprint arXiv:2403.17481, 2024

    Lu Lin and Ze Chen. A type of nonlinear Fréchet regressions.arXiv preprint arXiv:2403.17481, 2024

    work page arXiv 2024

  19. [19]

    Total variation regularized Fréchet regression for metric-space valued data.The Annals of Statistics, 49(6):3510–3533, 2021

    Zhenhua Lin and Hans-Georg Müller. Total variation regularized Fréchet regression for metric-space valued data.The Annals of Statistics, 49(6):3510–3533, 2021

    work page 2021

  20. [20]

    Convergence and trade-offs in Rie- mannian gradient descent and Riemannian proximal point.arXiv preprint arXiv:2403.10429, 2024

    David Martínez-Rubio, Christophe Roux, and Sebastian Pokutta. Convergence and trade-offs in Rie- mannian gradient descent and Riemannian proximal point.arXiv preprint arXiv:2403.10429, 2024. 25

    work page arXiv 2024

  21. [21]

    Ridge regression for manifold-valued time-series with application to meteo- rological forecast.arXiv preprint arXiv:2411.18339, 2024

    Esfandiar Nava-Yazdani. Ridge regression for manifold-valued time-series with application to meteo- rological forecast.arXiv preprint arXiv:2411.18339, 2024

    work page arXiv 2024

  22. [22]

    Fréchet regression for random objects with euclidean predictors.The Annals of Statistics, 47(2):691–719, 2019

    Alexander Petersen and Hans-Georg Müller. Fréchet regression for random objects with euclidean predictors.The Annals of Statistics, 47(2):691–719, 2019

    work page 2019

  23. [23]

    Convergence of nonsmooth descent methods via Kurdyka–Lojasiewicz inequality on Riemannian manifolds

    Seyedehsomayeh Hosseini. Convergence of nonsmooth descent methods via Kurdyka–Lojasiewicz inequality on Riemannian manifolds. November 2015

    work page 2015

  24. [24]

    Wasserstein geometry of Gaussian measures.Osaka Journal of Mathematics, 48(4):1005–1026, 2011

    Asuka Takatsu. Wasserstein geometry of Gaussian measures.Osaka Journal of Mathematics, 48(4):1005–1026, 2011

    work page 2011

  25. [25]

    Multivariate manifold-valued curve regression in time.arXiv preprint arXiv:2208.12585, 2022

    A Torres-Signes, MP Frías, and MD Ruiz-Medina. Multivariate manifold-valued curve regression in time.arXiv preprint arXiv:2208.12585, 2022

    work page arXiv 2022

  26. [26]

    Global minimization of a difference of two convex functions

    Hoang Tuy. Global minimization of a difference of two convex functions. InSelected Topics in Operations Research and Mathematical Economics: Proceedings of the 8th Symposium on Operations Research, Held at the University of Karlsruhe, West Germany August 22–25, 1983, pages 98–118. Springer, 1984

    work page 1983

  27. [27]

    Viaclovsky

    Jeff A. Viaclovsky. Math 865: Topics in Riemannian geometry.https://www.math.uci.edu /~jviaclov/courses/865_Fall_2011.pdf, 2011. Course notes

    work page 2011

  28. [28]

    Viaclovsky

    Jeff A. Viaclovsky. Critical metrics for Riemannian curvature functionals.https://www.math .uci.edu/~jviaclov/lecturenotes/PCMI_Lectures_Final.pdf, 2013. Expanded version of IAS/PCMI lecture notes, 2013

    work page 2013

  29. [29]

    On a class of geodesically convex optimization problems solved via Euclidean mm methods.arXiv preprint arXiv:2206.11426, 2022

    Melanie Weber and Suvrit Sra. On a class of geodesically convex optimization problems solved via Euclidean mm methods.arXiv preprint arXiv:2206.11426, 2022

    work page arXiv 2022

  30. [30]

    Ambient and intrinsic triangulations and topological methods in cosmology

    Mathijs Wintraecken. Ambient and intrinsic triangulations and topological methods in cosmology. 2015

    work page 2015

  31. [31]

    A globally convergent difference-of-convex algorithmic framework and application to log-determinant optimization problems.arXiv preprint arXiv:2306.02001, 2023

    Chaorui Yao and Xin Jiang. A globally convergent difference-of-convex algorithmic framework and application to log-determinant optimization problems.arXiv preprint arXiv:2306.02001, 2023

    work page arXiv 2023

  32. [32]

    Network regression with graph Laplacians.Journal of Machine Learning Research, 23(320):1–41, 2022

    Yidong Zhou and Hans-Georg Müller. Network regression with graph Laplacians.Journal of Machine Learning Research, 23(320):1–41, 2022

    work page 2022

  33. [33]

    Riemannian manifolds with positive sectional curvature

    Wolfgang Ziller. Riemannian manifolds with positive sectional curvature. InGeometry of Manifolds with Non-negative Sectional Curvature: Editors: Rafael Herrera, Luis Hernández-Lamoneda, pages 1–19. Springer, 2014. 26

    work page 2014