On the approximation of the Riemannian barycenter

Nina Miolane; P.-A. Absil; Simon Mataigne

REVIEW 2 major objections 5 minor 18 references

On the approximation of the Riemannian barycenter

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Minimizing a guaranteed lower bound of the Riemannian barycenter objective yields an approximate mean with a computable error certificate and, on the Stiefel manifold, a speedup of over 100 times.

desk verdict A clean certificate theorem undercut by an unstated gap between the exact-minimizer hypothesis and the approximate-minimizer algorithm. read the letter →

arxiv 2504.15671 v2 pith:ZMG3JMBN submitted 2025-04-22 math.DG

Simon Mataigne , P.-A. Absil , Nina Miolane This is my paper · ORCID

classification math.DG MSC 53-0853C22

keywords RiemannianbarycenterKarchermeanFréchetStiefelmanifolddistanceboundserrorcertificatelogarithmapproximateoptimization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Riemannian barycenters—manifold analogues of the mean—are usually found by gradient descent, which requires a Riemannian logarithm at every step and is intractable on manifolds such as the Stiefel manifold. This paper replaces the exact objective with a guaranteed lower-bound surrogate, minimizes the surrogate, and proves a computable upper bound on the relative error of the result. Experiments on the Stiefel manifold show the surrogate minimizer matches exact Riemannian gradient descent in accuracy while running more than a hundred times faster. The paper's numerical finding is that the certificate, though guaranteed, is much tighter for clustered datasets than for spread ones.

What carries the argument

The machinery is a sandwich: two functions $\underline{m}_g$ and $\overline{M}_g$ that bracket the Riemannian distance, $\underline{m}_g \le d_g \le \overline{M}_g$. Lifting these to the objective gives $\hat{f}_\nu \le f_\nu \le \hat{F}_\nu$. The key identity is the chain of inequalities in (6)—the surrogate minimizer beats the true minimizer on the surrogate, and the true minimizer beats the surrogate minimizer on the true objective—which yields Theorem 1's computable certificate. On the Stiefel manifold the explicit brackets $\underline{m}_\beta \le d_\beta \le \overline{M}_\beta$ from [12, Thm. 7.1] are used; because they involve only Euclidean distances, the surrogate can be optimized without computing a single Riemannian logarithm.

What would settle it

Compute the true geodesic distance for a pair of points on St$(n,p)$ with $n\ge 2p$ using a reliable numerical Riemannian logarithm and compare it with $\underline{m}_\beta$ and $\overline{M}_\beta$; a single pair where the lower bound exceeds the true distance or the upper bound falls below it would invalidate the certificate at the root of Theorem 1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: with under- and over-approximations of the Riemannian distance, letting $\hat{x}_*$ minimize the lower-bound surrogate $\hat{f}_\nu$ gives $$\frac{f_\nu(\hat{x}_*)-f_\nu(x_*)}{f_\nu(x_*)} \le \frac{\hat{F}_\nu(\hat{x}_*)-\hat{f}_\nu(\hat{x}_*)}{\hat{f}_\nu(\hat{x}_*)}.$$ The right-hand side contains only computable quantities, so the surrogate solution carries a rigorous, computable certificate of how far it is from the true barycenter. The paper further shows numerically that on the Stiefel manifold with the $\beta$-metric this surrogate minimization reaches essentially the same objective value as Riemannian gradient descent on the exact problem, that all computed certificates satisfy the inequality, and that the speedup exceeds a factor of 100 as the dimension grows.

Load-bearing premise

The whole construction stands on the proven bracketing inequalities $\underline{m}_\beta \le d_\beta \le \overline{M}_\beta$ being valid for every pair of points on the Stiefel manifold with $n\ge 2p$; if any pair violated them, the chain in (6) would break and Theorem 1 would no longer certify the approximation.

Editorial extensions

If this is right

On any manifold with proven distance brackets, a user can minimize the lower-bound surrogate and obtain a rigorous, computable upper bound on the relative error of the resulting approximate barycenter without computing a Riemannian logarithm.
On the Stiefel manifold with the $\beta$-metric, the surrogate minimization matches exact Riemannian gradient descent in objective value while running more than 100 times faster; the lifting-map baseline has similar cost but lacks guaranteed behavior on spread datasets, whereas the surrogate carries the Theorem 1 certificate.
The certificate is always satisfied in the reported experiments and is tighter for clustered datasets, where the distance brackets are sharper.
The same sandwich construction yields a computable upper bound on the geodesic distance between the true and approximate barycenters (Theorem 2).
Riemannian gradient descent on the exact objective becomes intractable for large $N,n,p$, while the surrogate approach remains practical, making it a worthwhile alternative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the certificate could be used as an online stopping rule for surrogate optimization: iterate until the computable bracket gap $\hat{F}_\nu(\hat{x})-\hat{f}_\nu(\hat{x})$ falls below a desired tolerance.
Beyond the paper, any manifold equipped with provable global distance brackets—for example spheres, flag manifolds, or fixed-rank matrices—inherits the same approximate-barycenter scheme and certificate without new algorithmic machinery.
Beyond the paper, one could compare the two surrogate choices directly: minimizing the upper bound $\hat{F}_\nu$ rather than $\hat{f}_\nu$ minimizes the Theorem 2 distance bound, and may produce smaller geodesic error even though its relative objective certificate is weaker.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Simon,

The theorem is fine; the paper's central numerical claim isn't connected to it. That's the short version.

What's new: the paper proposes bounding the Riemannian distance from below and above, then minimizing the lower-bound surrogate, with Theorem 1 giving a computable relative-error certificate. The chain (6) is correct under the stated hypothesis that bx* exactly minimizes bfν. Theorems 2 and 3 are also correct and elementary. As a contribution to manifold statistics, this is a sensible and useful idea, and the Stiefel experiments show that the surrogate (A2) can be over two orders of magnitude faster than exact Riemannian gradient descent (A1). That's worth something.

The soft spot is the assumption gap. The algorithm actually run for (A2) stops when the relative change in bfν is below 1e-6. That is a stationarity heuristic, not a global optimality certificate. The first inequality in the chain, bfν(Y*) ≤ bfν(x*), only holds if Y* is a true global minimizer of bfν. Without that, the quantity reported in Table 1 as an upper bound on the relative error is not proven to be an upper bound. The paper's conclusion that (A2) 'benefits from theoretical guarantees (Theorem 1)' overstates what the experiment supports. This is the one load-bearing flaw in the numerical narrative, and the paper doesn't acknowledge it.

The other complaints are minor: no error bars or seeds on the 10 runs, no commit hash for the repository. Those are standard hygiene and easy to fix. The distance bounds from [12] are cited, not re-derived; that's fine, though a reader would want to check [12] for the Stiefel-specific inequalities.

Who is this for? People who compute barycenters on Stiefel or similar manifolds and want a certified surrogate. The idea is good enough that I'd send it out, but the revision has to either prove the surrogate has no spurious local minima (unlikely for a general manifold), switch to a global optimization method that can certify the minimum, or soften the claims to say the bound is heuristic on the returned point. As is, the certificate is real but the algorithm does not earn it.

Recommendation: accept for review with a clear demand to fix the minimizer gap and tighten the numerics. The core math is sound; the packaging needs work.

— [Your name]

Referee Report

2 major / 5 minor

Summary. This manuscript introduces a framework for approximating the Riemannian lν-barycenter by minimizing a surrogate objective built from certified lower and upper bounds on the Riemannian distance. For an exact global minimizer bx* of the lower-bound surrogate bfν, Theorem 1 provides a computable relative-error certificate for fν, and Theorem 2 provides a computable distance upper bound; Theorem 3 gives a lower bound that depends on the unknown optimal value. The method is instantiated on the Stiefel manifold using distance bounds from the authors' prior work [12], and Section 5 compares three algorithms: Riemannian gradient descent on the exact objective, RGD on the lower-bound surrogate, and a lifting-map fixed-point iteration. The experiments report accurate approximations, satisfaction of the bounds, and a speedup exceeding 100× for surrogate-based methods. The central theoretical result is elementary and correct, but the numerical section does not instantiate the exact-global-minimizer hypothesis of Theorem 1.

Significance. If the claimed certificate could be attached to the actual algorithm, the paper would contribute a useful and inexpensive error-controlled alternative to exact Riemannian barycenter computation on manifolds with costly logarithms. Strengths include the generic bracketing-function formulation, the simple and correct proofs of Theorems 1–3, the concrete Stiefel bounds inherited from [12], and the public code. The main limitation is that the implemented Algorithm (A2) does not satisfy the hypothesis of Theorem 1, so the strongest advertised guarantee is not currently supported; this limits the significance until the gap is closed or the claim is appropriately qualified.

major comments (2)

[Section 5, Eq. (7), Table 1] Theorem 1 assumes bx* ∈ argmin_{x∈M} bfν(x). The algorithm (A2) minimizes bfν^ν by Riemannian gradient descent and stops when |bfν(Y_k)^ν − bfν(Y_{k−1})^ν| ≤ ε bfν(Y_0)^ν. This stopping rule is a stationarity/fixed-point criterion; it does not imply bfν(Y*) ≤ bfν(x*) and does not imply global optimality. Consequently the first inequality in chain (6), which is needed for the certificate, is not established for the reported Y*. The quantities in Table 1 of the form [bFν(Y*) − bfν(Y*)] / bfν(Y*) are therefore not proven upper bounds on the true relative error [fν(Y*) − fν(x*)] / fν(x*). The concluding sentence that (A2) “benefits from theoretical guarantees (Theorem 1)” should be qualified; the authors should either establish a global optimality property for bfν on St(n,p), add a stationarity-gap correction to the bound, or present the table as a heuristic validation.
[Section 5, Table 1] The left-hand side [fν(X*) − fν(Y*)] / fν(X*) in Table 1 uses X* produced by A1 as a proxy for the true barycenter x*. A1 is itself only run to a stationarity criterion and its Riemannian logarithm is numerically approximated. Thus the tabulated relative error is an empirical comparison against an approximate reference, not a measured distance to x*. Please state this limitation explicitly and, if possible, report the A1 tolerance and confirm that the conclusions are insensitive to it.

minor comments (5)

[Table 1] Averages over 10 runs are reported without standard deviations or confidence intervals; given that the speed and accuracy claims are central, the variability should be quantified.
[Section 5] The stopping criterion notation should be written with an explicit iteration index, for example |fν(X_k^*)^ν − fν(X_{k−1}^*)^ν| ≤ ε fν(X_0)^ν, to remove the ambiguity in the current “X∗−1” notation.
[Section 3] The formulas for bmβ and cMβ are used with β = 1/2 in the experiments, but the β-metric itself is not defined in this paper; please define it or give a precise pointer to [12, Eq. (2.1)].
[Section 5] The sampling procedures for “spread” and “clustered” datasets are described informally; specify the exact distributions and the injectivity-radius procedure used so the experiments can be reproduced from the text alone.
[Figure 3, bottom] The running-time comparison reports a speedup larger than 100× but gives no measurement details; please state the hardware, software versions, and use median timings with error bars rather than a single benchmark curve.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is a generic bracketing sandwich; the cited Stiefel bounds are parameter-free and independent of barycenter quantities.

full rationale

The central theoretical result, Theorem 1, is a generic inequality over arbitrary bracketing functions bm_g ≤ d_g ≤ cM_g. The proof is the sandwich chain (6): bfν(bx*) ≤ bfν(x*) ≤ fν(x*) ≤ fν(bx*) ≤ bFν(bx*). Each inequality follows from (i) bx* minimizing bfν, (ii) x* minimizing fν, and (iii) pointwise distance bounds (5). No parameter is fitted to the data being predicted, and the bound (7) is not equivalent to its inputs by construction; it is a nontrivial consequence of the sandwich. The Stiefel instantiation imports explicit distance bounds bmβ and cMβ from the authors' prior work [12, Thm. 7.1 and Cor. 7.2]. That is a self-citation, but the cited result is parameter-free, has stated assumptions (β-metric, n ≥ 2p) that do not involve any barycenter quantity, and is independently checkable; hence it constitutes real evidence, not a circular premise. The paper also cites [13] and [2] by the same authors for the Riemannian logarithm and the injectivity radius, but these are auxiliary technical inputs rather than assumptions equivalent to the barycenter approximation claim. One non-circular but real gap: the experiments stop (A2) by an RGD criterion on bfν, which does not guarantee that the returned Y* is an exact global minimizer of bfν. Therefore the inequality bfν(Y*) ≤ bfν(x*) required for Theorem 1 is not rigorously established numerically, and Table 1's reported bound is a heuristic certificate rather than a proven one. This is a correctness/application gap, not a circular-reasoning defect; the theorem itself remains a self-contained derivation.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central theory is generic: any bracketing functions bm and cM yield Theorems 1 to 3. The numerical claims additionally import the Stiefel distance bounds from the authors' earlier preprint [12], plus standard existence and differentiability results from [3,4]. No new entities are introduced. The only hand-chosen experimental knob is the stopping tolerance.

free parameters (1)

Stopping tolerance epsilon = 1e-6
Hand-chosen in Section 5 to stop algorithms (A1), (A2) and (A3). It affects the reported iterates and table values but not the theoretical theorems.

assumptions (3)

domain assumption Stiefel distance bounds bmβ <= dβ <= cMβ from [12, Thm. 7.1]
Enter Section 3 as the instantiation of (5). The current paper does not prove these bounds; it assumes n >= 2p and the β-metric.
domain assumption Existence and uniqueness conditions for the Riemannian lν-barycenter, and the gradient formula grad dg(x,xi)^ν = -ν dg(x,xi)^(ν-2) Log_x(xi) away from cut loci
Invoked around equations (1) and (4), via references [3] and [4]. This is required for x* to be well defined and for the stationarity condition.
standard math Hopf-Rinow completeness and zero-measure cut locus for the Riemannian logarithm
Section 2 background; needed so that dg is realized by geodesics and Log_x is single-valued almost everywhere.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the approximation of the Riemannian barycenter." pith.science (2026). https://pith.science/paper/ZMG3JMBN

@misc{pith2026250415671,
  author       = {Pith},
  title        = {Pith review of: On the approximation of the Riemannian barycenter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMG3JMBN}},
  note         = {Machine review of arXiv:2504.15671}
}

read the original abstract

We present a method for computing an approximate Riemannian barycenter of a collection of points lying on a Riemannian manifold. Our approach relies on the use of theoretically proven under- and over-approximations of the Riemannian distance function. We compare it to Riemannian steepest descent on the exact objective function of the Riemannian barycenter and to an approach that approximates the Riemannian logarithm using lifting maps. Experiments are conducted on the Stiefel manifold.

Figures

Figures reproduced from arXiv: 2504.15671 by the authors.

**Figure 1.** An artist view of the Riemannian l ν -barycenter. The figure represents a collection of points {xi} 3 i=1 ⊂ M and their Riemannian l ν -barycenter x ∗ . If ν ≥ 2, x ∗ must be a first-order stationary point of (1), i.e., satisfies (4). 3 Minorizing and majorizing the objective function Assume one knows two functions mb g, Mcg : M2 → R such that for all pairs x, xe ∈ M, we have mb g(x, xe) ≤ dg(x, xe) ≤ Mcg(x, xe). (… view at source ↗

**Figure 2.** An additional caveat is the difficulty to define the mapping Lx(xe) for all xe ∈ M. For example, on the Stiefel manifold, the inverses of the polar retraction, the QR retraction and the Cayley retraction have guaranteed existence only in a neighborhood of x [10]. Logx (x2) Logx (x1) ProjTxM(x2 − x) ProjTxM(x1 − x) M x 0x TxM x1 x2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** At the top: convergence curves of algorithms (A1), (A2) and (A3) to obtain the results of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 6 canonical work pages

[12]

Mataigne, S., Absil, P.-A., Miolane, N.: Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics (2024),https://arxiv.org/ abs/2408.07072

arXiv 2024
[1]

Princeton University Press, Princeton, NJ (2008), http://press

Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2008), http://press. princeton.edu/titles/8586.html

work page 2008
[2]

SIAM Journal on Matrix Analysis and Applications46(2), 1145–1167 (2025), https://doi.org/10.1137/24M1644808

Absil, P.-A., Mataigne, S.: The Ultimate Upper Bound on the Injectivity Radius of the Stiefel Manifold. SIAM Journal on Matrix Analysis and Applications46(2), 1145–1167 (2025), https://doi.org/10.1137/24M1644808

doi:10.1137/24m1644808 2025
[3]

Proceedings of the American Mathematical Society139(2), 655–673 (2011),http: //www.jstor.org/stable/41059320

Afsari, B.: RiemannianLp center of mass: Existence, uniqueness, and convexity. Proceedings of the American Mathematical Society139(2), 655–673 (2011),http: //www.jstor.org/stable/41059320

work page arXiv 2011
[4]

SIAM Journal on Control and Optimization51(3), 2230–2260 (2013), https://doi.org/10.1137/12086282X

Afsari, B., Tron, R., Vidal, R.: On the convergence of gradient descent for finding the Riemannian center of mass. SIAM Journal on Control and Optimization51(3), 2230–2260 (2013), https://doi.org/10.1137/12086282X

doi:10.1137/12086282x 2013
[5]

Foundations of Computational Mathematics23(4), 1335–1380 (2023), https://doi.org/10.1007/s10208-022-09571-x

Barbero-Liñán, M., de Diego, D.M.: Retraction maps: A seed of geometric in- tegrators. Foundations of Computational Mathematics23(4), 1335–1380 (2023), https://doi.org/10.1007/s10208-022-09571-x

doi:10.1007/s10208-022-09571-x 2023
[6]

Bouchard, F., Laurent, N., Said, S., Le Bihan, N.: Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds (2025), https: //arxiv.org/abs/2501.11555

work page Pith review arXiv 2025
[7]

The Annals of Statistics47(1), 415 – 438 (2019),https://doi.org/ 10.1214/18-AOS1692

Chakraborty, R., Vemuri, B.C.: Statistics on the Stiefel manifold: Theory and applications. The Annals of Statistics47(1), 415 – 438 (2019),https://doi.org/ 10.1214/18-AOS1692

doi:10.1214/18-aos1692 2019

Show all 18 references

[8]

SIAM Journal on Matrix Analysis and Applications20(2), 303–353 (1998), https://doi.org/10.1137/S0895479895290954

Edelman, A., Arias, T.A., Smith, S.T.: The Geometry of Algorithms with Orthog- onality Constraints. SIAM Journal on Matrix Analysis and Applications20(2), 303–353 (1998), https://doi.org/10.1137/S0895479895290954

1998 doi
[9]

IEEE Transactions on Signal Processing 63(1), 155–168 (2015),https://doi.org/10.1109/TSP.2014.2365764

Fiori, S., Kaneko, T., Tanaka, T.: Tangent-bundle maps on the Grassmann mani- fold: Application to empirical arithmetic averaging. IEEE Transactions on Signal Processing 63(1), 155–168 (2015),https://doi.org/10.1109/TSP.2014.2365764

2015
[10]

IEEE Transactions on Signal Processing61, 883–894 (02 2013), https://doi.org/10.1109/TSP.2012.2226167

Kaneko, T., Fiori, S., Tanaka, T.: Empirical arithmetic averaging over the compact Stiefel manifold. IEEE Transactions on Signal Processing61, 883–894 (02 2013), https://doi.org/10.1109/TSP.2012.2226167

2013
[11]

Karcher, H.: Riemannian Center of Mass and so called Karcher mean (2014), https://arxiv.org/abs/1407.2087

2014 arXiv
[13]

SIAM Journal on Matrix Analysis and Applications46(2), 879–905 (2025), https://doi.org/10.1137/24M1647801

Mataigne, S., Zimmermann, R., Miolane, N.: An Efficient Algorithm for the Rie- mannian Logarithm on the Stiefel Manifold for a Family of Riemannian Met- rics. SIAM Journal on Matrix Analysis and Applications46(2), 879–905 (2025), https://doi.org/10.1137/24M1647801

2025 doi
[14]

Journal of Mathematical Imaging and Vision25(1), 127–154 (2006), https://doi.org/10.1007/s10851-006-6228-4, a preliminary appeared as INRIA RR-5093, January 2004

Pennec, X.: Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geomet- ric Measurements. Journal of Mathematical Imaging and Vision25(1), 127–154 (2006), https://doi.org/10.1007/s10851-006-6228-4, a preliminary appeared as INRIA RR-5093, January 2004

2006 doi
[15]

Fields Institute Communications, Amer- ican Mathematical Society (1996), https://books.google.be/books?id= ODDyngEACAAJ

Sakai, T.: Riemannian Geometry. Fields Institute Communications, Amer- ican Mathematical Society (1996), https://books.google.be/books?id= ODDyngEACAAJ

1996
[16]

Computational Optimization and Applications 77(3), 779–810 (2020), https:// doi.org/10.1007/s10589-020-00219-6 10 Simon Mataigne, P.-A

Zhu, X., Sato, H.: Riemannian conjugate gradient methods with inverse retraction. Computational Optimization and Applications 77(3), 779–810 (2020), https:// doi.org/10.1007/s10589-020-00219-6 10 Simon Mataigne, P.-A. Absil and Nina Miolane

2020 doi
[17]

SIAM Journal on Matrix Analysis and Applications 43(2), 953–980 (2022),https://doi.org/10.1137/21M1425426

Zimmermann, R., Hüper, K.: Computing the Riemannian logarithm on the Stiefel manifold: Metrics, methods, and performance. SIAM Journal on Matrix Analysis and Applications 43(2), 953–980 (2022),https://doi.org/10.1137/21M1425426

2022 doi
[18]

SIAM Journal on Matrix Analysis and Applications 46(1), 298–309 (2025),https://doi.org/10.1137/24M1663818

Zimmermann, R., Stoye, J.: The Injectivity Radius of the Compact Stiefel Manifold under the Euclidean Metric. SIAM Journal on Matrix Analysis and Applications 46(1), 298–309 (2025),https://doi.org/10.1137/24M1663818

2025 doi

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [12]

Mataigne, S., Absil, P.-A., Miolane, N.: Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics (2024),https://arxiv.org/ abs/2408.07072

arXiv 2024

[2] [1]

Princeton University Press, Princeton, NJ (2008), http://press

Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2008), http://press. princeton.edu/titles/8586.html

work page 2008

[3] [2]

SIAM Journal on Matrix Analysis and Applications46(2), 1145–1167 (2025), https://doi.org/10.1137/24M1644808

Absil, P.-A., Mataigne, S.: The Ultimate Upper Bound on the Injectivity Radius of the Stiefel Manifold. SIAM Journal on Matrix Analysis and Applications46(2), 1145–1167 (2025), https://doi.org/10.1137/24M1644808

doi:10.1137/24m1644808 2025

[4] [3]

Proceedings of the American Mathematical Society139(2), 655–673 (2011),http: //www.jstor.org/stable/41059320

Afsari, B.: RiemannianLp center of mass: Existence, uniqueness, and convexity. Proceedings of the American Mathematical Society139(2), 655–673 (2011),http: //www.jstor.org/stable/41059320

work page arXiv 2011

[5] [4]

SIAM Journal on Control and Optimization51(3), 2230–2260 (2013), https://doi.org/10.1137/12086282X

Afsari, B., Tron, R., Vidal, R.: On the convergence of gradient descent for finding the Riemannian center of mass. SIAM Journal on Control and Optimization51(3), 2230–2260 (2013), https://doi.org/10.1137/12086282X

doi:10.1137/12086282x 2013

[6] [5]

Foundations of Computational Mathematics23(4), 1335–1380 (2023), https://doi.org/10.1007/s10208-022-09571-x

Barbero-Liñán, M., de Diego, D.M.: Retraction maps: A seed of geometric in- tegrators. Foundations of Computational Mathematics23(4), 1335–1380 (2023), https://doi.org/10.1007/s10208-022-09571-x

doi:10.1007/s10208-022-09571-x 2023

[7] [6]

Bouchard, F., Laurent, N., Said, S., Le Bihan, N.: Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds (2025), https: //arxiv.org/abs/2501.11555

work page Pith review arXiv 2025

[8] [7]

The Annals of Statistics47(1), 415 – 438 (2019),https://doi.org/ 10.1214/18-AOS1692

Chakraborty, R., Vemuri, B.C.: Statistics on the Stiefel manifold: Theory and applications. The Annals of Statistics47(1), 415 – 438 (2019),https://doi.org/ 10.1214/18-AOS1692

doi:10.1214/18-aos1692 2019