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Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The RL-barycenter of manifold-valued data is the Euclidean projection of the arithmetic mean onto the manifold, so Stiefel and Grassmann averaging collapses to a single matrix projection.

desk verdict The projected mean is a real RL-barycenter, but it is not the only one; Proposition 1's uniqueness claim is too strong. read the letter →

arxiv 2501.11555 v1 pith:Q2WIMMQ3 submitted 2025-01-20 stat.ML cs.LG

Florent Bouchard , Nils Laurent , Salem Said , Nicolas Le Bihan This is my paper · ORCID
classification stat.MLcs.LG MSC 62R30
keywords RL-barycenterStiefelmanifoldGrassmannprojectedarithmeticmeanretractionliftingpolardecompositioneigenprojector
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 averaging normally requires exponential and logarithm maps that are expensive or not available in closed form. This paper proposes a simpler averaging rule, the RL-barycenter, which replaces those maps by a retraction and a lifting inside a fixed-point equation. Its central result is that, with the natural projection retraction and tangent-space lifting, the RL-barycenter equals the Euclidean projection of the arithmetic mean onto the manifold. On Stiefel manifolds that projection is the polar factor of the mean matrix; on Grassmann manifolds it is the top-$k$ eigenprojector. Iterative mean computations can therefore be replaced by one matrix decomposition.

What carries the argument

The central mechanism is the RL-barycenter fixed-point equation, $G=R_G\left(\frac{1}{n}\sum_{i=1}^n L_G(M_i)\right)$, where a retraction maps tangent vectors onto the manifold and a lifting maps manifold points back into the tangent space. Choosing the closest-point retraction $R_G(\xi)=P(G+\xi)$ and the tangent-space lifting $L_G(M)=P_G(M-G)$ makes the fixed-point condition $\frac{1}{n}\sum_i P_G(M_i-G)=0$ coincide with the first-order optimality condition of the Euclidean projection $P(\frac{1}{n}\sum_i M_i)$, so the projected mean is the fixed point. On Stiefel, $P$ is the polar factor $\mathrm{uf}(\cdot)$; on Grassmann, $P$ is the top-$k$ eigenprojector; for the QR variant, the lifting is the differential $d\,\mathrm{qf}(G)[\cdot]$, whose vanishing at $\mathrm{qf}(\frac{1}{n}\sum_i M_i)$ is verified in the supplementary material.

What would settle it

Take two antipodal unit vectors on the circle, which is the Stiefel manifold $\mathrm{St}(2,1)$: their arithmetic mean is zero, the polar projection is not unique, and every point of the circle satisfies the RL fixed-point equation, so no unique projected mean exists; a Grassmann analogue occurs when the mean matrix has equal $k$-th and $(k+1)$-th eigenvalues.

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Extended reading notes

Core claim

Proposition 1 states that if the retraction is $R_G(\xi)=P(G+\xi)$, where $P$ is the closest-point projection onto the manifold, and the lifting is $L_G(M)=P_G(M-G)$, the orthogonal projection of $M-G$ onto the tangent space at $G$, then the RL-barycenter of samples $\{M_i\}_{i=1}^n$ is $G=P\left(\frac{1}{n}\sum_{i=1}^n M_i\right)$. On the Stiefel manifold this projection is the polar factor $\mathrm{uf}(\frac{1}{n}\sum_i M_i)$, and the paper observes that this same point is a closed-form solution of the R-barycenter based on the orthographic retraction. On the Grassmann manifold the projection sends a symmetric matrix to the eigenprojector onto its $k$ dominant eigenvectors. The supplementary material proves the QR-based projection $\mathrm{qf}(\frac{1}{n}\sum_i M_i)$ is also an RL-barycenter when the retraction is QR and the lifting is the differential $d\,\mathrm{qf}(G)[M-G]$.

Load-bearing premise

The projected-mean formula requires the closest-point projection onto the manifold to be unique at the arithmetic mean: for Stiefel, the mean matrix must have full column rank so the polar factor is unique, and for Grassmann, the $k$-th and $(k+1)$-th largest eigenvalues must be distinct.

Editorial extensions

If this is right

  • When the central claim holds, the Stiefel R-barycenter based on the orthographic retraction is obtained in closed form by one polar decomposition of the arithmetic mean, with no iterative fixed-point loop.
  • On Grassmann, averaging subspaces reduces to an eigendecomposition of the arithmetic mean matrix, bypassing Riemannian logarithms and their iterative solvers.
  • The projected-mean estimator is non-iterative and uses only standard matrix decompositions, making it cheaper than Fréchet-mean or R-barycenter iterations on the same manifolds.
  • The simulations show the projected mean competing with R-barycenters on Stiefel and staying close to the Riemannian mean on Grassmann at the tested noise levels.

Reading between the lines

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

  • Editorial inference: the same projection shortcut should extend to other manifolds with computable closest-point projections, such as fixed-rank positive semidefinite matrices, but the single-valuedness conditions would need to be checked for each manifold.
  • Editorial inference: the formula provides a cheap plug-in mean for subspace-based clustering and dimensionality-reduction pipelines, a use not tested in the paper.
  • Editorial inference: on degenerate or small-sample data the projection can become multivalued, so practical implementations need a tie-breaking or regularization rule; the paper does not address this regime.
  • Editorial inference: a natural follow-up experiment is to measure how the projected mean's bias relative to the Fréchet mean grows with sample dispersion on Stiefel, which would delimit where the shortcut is a good approximation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes RL-barycenters, an averaging method on matrix manifolds that replaces the inverse retraction in R-barycenters with a simpler lifting. The main result (Proposition 1) states that, for the projection retraction and the orthogonal-projection lifting, the RL-barycenter of samples {M_i} is the projection onto the manifold of their arithmetic mean. The authors apply this to the Stiefel manifold with polar and QR projections and to the Grassmann manifold, for which they derive the ambient-space projection (Proposition 2). Numerical experiments compare the resulting projected means with R-barycenters and the Riemannian mean on simulated data, and the paper reports competitive accuracy at lower computational cost. Code is provided for reproducibility.

Significance. If Proposition 1 is read as stating that the projected arithmetic mean is an RL-barycenter, the paper gives a clean, simple, and computationally cheap averaging procedure for Stiefel and Grassmann manifolds. The closed-form connection to the orthographic R-barycenter on Stiefel is a useful observation, and the Grassmann projection in Proposition 2 is a correct and practical tool. The QR-extension proof in the supplementary materials is a genuine derivation, and the availability of code is a strength. However, the paper's stronger reading of Proposition 1 as uniqueness is false: the RL-barycenter fixed-point equation has multiple solutions, and the proof establishes only sufficiency. This is a load-bearing correctness issue that must be fixed before the main claims in the abstract and introduction are accurate.

major comments (3)
  1. [Section III, Definition 1 and Proposition 1] The statement that the RL-barycenter 'is' P(1/n Σ M_i) is a uniqueness claim, but Definition 1's fixed-point equation generally has many solutions. Because the projection retraction satisfies R_G(0)=G for every G, any G with (1/n)Σ_i L_G(M_i)=0 is an RL-barycenter, and the proof of Proposition 1 only shows that G=P(A) has this property. This is not a degenerate edge case. For example, on Gr_{3,1}, take 16 copies of e1e1^T, 3 copies of e2e2^T, and 1 copy of e3e3^T; then A=diag(0.8,0.15,0.05), P(A)=e1e1^T, but G'=e3e3^T also satisfies P_{G'}(A-G')=2 sym((I-G')(A-G')G')=0, so both are RL-barycenters. Similarly, on St_{2,1}, for M1=e1 and M2=e2, both (e1+e2)/√2 and -(e1+e2)/√2 are RL-barycenters. Proposition 1 should be rephrased as 'P(A) is an RL-barycenter', and the abstract, introduction, and the closed-form claim about the orthographic R-barycenter should say 'a closed-form solution' rather than 'the barycenter'.
  2. [Section III, Proposition 1] The formula G=P(A) requires that the projection onto the manifold be single-valued at A, but no such condition is stated. For the Stiefel manifold, the polar factor uf(A) is unique only when A has full column rank; for the Grassmann manifold, the eigenspace of the k largest eigenvalues is unique only when λ_k(A)>λ_{k+1}(A). These assumptions can fail on small or degenerate samples, in which case 'the' projected mean is not defined. For instance, on St_{2,1} with samples e1 and -e1, the arithmetic mean is 0 and has no polar factor; on Gr_{2,1} with samples e1e1^T and e2e2^T, the mean has equal eigenvalues and any one-dimensional subspace is a projection. The theorem should either state these conditions explicitly or be qualified to samples in general position. The same caveat applies to the QR-based projected mean in the supplementary materials, where qf(A) requires a unique QR factorization.
  3. [Section IV] The numerical comparison against existing R-barycenters is not fully reproducible as written. The paper reports results for 'R polar' and 'R QR' but does not specify the iterative algorithm used to solve the fixed-point equation (12), the stopping criterion, the tolerance, the maximum number of iterations, or how the inverse retractions were computed for the polar and QR retractions. Since Figure 1 is used to support the claim that the proposed projected means perform better than existing R-barycenters at larger noise levels, these implementation details are needed to rule out that the observed behavior is an artifact of a poorly tuned solver. Please add them or state explicitly that the comparison follows the implementation of [17].
minor comments (5)
  1. [Abstract and Introduction] The phrases 'the resulting RL-barycenter is ...' and 'our framework yields ... barycenters' should be adjusted to reflect the corrected statement that the projected arithmetic mean is an RL-barycenter, not necessarily the unique one.
  2. [Section III, after Proposition 1] The sentence 'G = uf(1/n Σ M_i) is a closed form solution for the R-barycenter with the orthographic retraction' should read 'a closed form solution', since non-uniqueness is also possible in that setting.
  3. [Supplementary Materials B] The definition of tril is ambiguous: it is used to mean the strictly lower triangular part ('cancels the diagonal and upper triangular elements'), but standard notation often includes the diagonal. Please state explicitly that tril returns only the strictly lower triangular part, because the proof of d qf(G)[A-G]=0 relies on this convention.
  4. [References / reproducibility] The GitHub URL in the Introduction appears as 'https://github.com/flbouchard/projection barycenter' with a space; the link should be checked and corrected so that the code is actually accessible.
  5. [Section II, notation] The notation for the retractions R^uf_U, R^qf_U, and R^o^{-1}_U is a little dense; a short sentence clarifying that the superscripts identify the retraction type would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Proposition 1 is a direct theorem from the chosen retraction and lifting, not a fit or self-citation chain.

full rationale

The paper's central claim is that the RL-barycenter for the metric-projection retraction (16) and tangent-projection lifting (17) is the Euclidean projection of the arithmetic mean. Proposition 1 proves this by showing that the projected mean has zero average lifting, using the first-order optimality condition of the projection; no empirical constant is fitted and no datum-dependent quantity is inserted into the argument. The retraction and lifting are chosen as simple natural operators and are not defined in terms of the projected mean, so the conclusion is not contained in the definition. The only prior results used as load-bearing are external references for standard retractions and projection-like maps ([17]–[19]); the minor self-citations ([7], [14], [27]) are background applications and do not support the main theorem. The proof's reliance on a sufficient condition for the fixed-point equation means the existence of other solutions (non-uniqueness) is a possible rigor gap, but this is a correctness issue, not circularity: the paper does not assume the projected mean is the barycenter in order to prove it. No equation is equivalent to its input by construction, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a clean but narrow setup: Euclidean embedded submanifolds with projection retractions and orthogonal-projection liftings. No free parameters are fitted, and no new entities are postulated. The main unstated assumption is the non-degeneracy of the arithmetic mean so that the projection is a single point.

assumptions (4)
  • domain assumption The manifold M is embedded in a Euclidean space E and the projection P(X) = argmin_{G in M} ||X-G||^2 is used as the retraction.
    This is the setup in Section III that yields the projected mean result. It holds for Stiefel and Grassmann with the Euclidean metric, but is only one of many possible retractions.
  • domain assumption The projection P is single-valued and the arithmetic mean of the data is in its smoothness domain.
    Needed for the point G = P(mean) to be well-defined. Fails if the mean matrix is rank-deficient (Stiefel) or has tied eigenvalues at the truncation threshold (Grassmann); the paper does not discuss this.
  • standard math The orthogonal projection onto the tangent space, P_G, is the differential of the retraction at G.
    From Absil and Malick (2012) for projection-like retractions; used to connect the lifting with the derivative of qf in the QR case.
  • standard math The Riemannian gradient of the squared distance to a fixed point is the tangential component of the Euclidean gradient.
    Standard result for submanifolds with the induced Euclidean metric, used in the proof of Proposition 1.

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Pith. "Pith review of Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds." pith.science (2026). https://pith.science/paper/Q2WIMMQ3

@misc{pith2026250111555,
  author       = {Pith},
  title        = {Pith review of: Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2WIMMQ3}},
  note         = {Machine review of arXiv:2501.11555}
}
read the original abstract

In this paper, the issue of averaging data on a manifold is addressed. While the Fr\'echet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this, R-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However, R-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call RL-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.

Figures

Figures reproduced from arXiv: 2501.11555 by the authors.

Figure 1
Figure 1. Medians (solid lines), 10% and 90% quantiles (filled areas) over [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Medians (solid lines), 10% and 90% quantiles (filled areas) over [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the approximation of the Riemannian barycenter

    math.DG 2025-04 conditional novelty 5.0 of 10

    Minimizing a proven lower bound on the Riemannian distance yields a certified, logarithm-free approximation of the Riemannian barycenter, demonstrated on the Stiefel manifold.

Reference graph

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    Hence, ∇f (V k) =0, and V k is a critical point

    It follows that ∇f (V k) = V p−k 0 I p−k I k − Λk 0 0 I p−k − Λp−k I k 0 . Hence, ∇f (V k) =0, and V k is a critical point. It remains to determine the set of k eigenvectors of X which yields the minimum. To do so, let’s look at the cost function at V k, which is f (V k) =∥X −...

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