{"id":"9652d87c-7fa8-45e5-a9ce-e543c14fddd2","arxiv_id":"2501.11555","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"On Stiefel and Grassmann manifolds, the RL-barycenter with a projection retraction and a tangent-space lifting is exactly the projected arithmetic mean, giving a closed-form low-cost averaging method.","lead":"A new averaging rule for data on curved spaces such as Stiefel and Grassmann manifolds reduces to the ordinary arithmetic mean followed by a projection back onto the manifold. This gives a cheap, closed-form alternative to expensive iterative geometric averaging methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1 states 'the' RL-barycenter equals the projected mean, but the fixed-point equation has many solutions even in nondegenerate cases; the proof only shows the projected mean is one RL-barycenter.","rationale":"The paper's practical contribution, the projected arithmetic mean as a cheap and competitive estimator, survives the critique: the algebraic computations for Stiefel and Grassmann are internally consistent, and the numerical experiments support the method. The load-bearing weakness is theoretical: Definition 1 does not select the projected mean uniquely, and the non-uniqueness is not confined to the degenerate cases flagged by the reader. On Grassmann, any spectral projector onto k eigenvectors of the arithmetic mean is an RL-barycenter; on Stiefel, sign-flipped projections are additional solutions. Therefore the headline theorem is overstated as written. This does not warrant rejection, but it warrants conditional acceptance: Proposition 1 should be corrected to assert existence, and the paper should either embrace non-uniqueness or add a canonical selection rule such as global minimization of the Euclidean distance to define 'the' RL-barycenter. With that revision, the mathematical content and the algorithm stand.","tokens_in":9510,"tokens_out":17459,"duration_ms":201171,"concrete_test":"Check the Gr_{3,1} counterexample: build the 20-sample set above, compute A = diag(0.8, 0.15, 0.05), and evaluate the lifting sum in Definition 1 for G = e1e1^T and G' = e3e3^T using Eq. (9). Both sums should be zero, and R_G(0) = G, R_{G'}(0) = G', so both are RL-barycenters. If confirmed, Proposition 1 must be rephrased as an existence result (the projected mean is an RL-barycenter), with uniqueness conditions or a selection rule stated explicitly.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 1 (Sec. III) proves that for G = P(A), the mean lifting is zero, so G satisfies Definition 1. This is only a sufficient condition. Because R_G is the metric projection, R_G(0) = G for every G on the manifold, so any G with zero mean lifting is an RL-barycenter. The zero-mean-lifting equation has many solutions. Example on Gr_{3,1}: take 20 samples with 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) and the projected mean is G = e1e1^T. But G' = e3e3^T also satisfies P_{G'}(A - G') = 2 sym((I - G')(A - G')G') = 0, so the mean lifting is zero and Definition 1 is satisfied. Thus both G and G' are RL-barycenters. The same phenomenon occurs on St_{2,1}: for M1 = e1, M2 = e2, both (e1+e2)/sqrt(2) and -(e1+e2)/sqrt(2) are fixed points. Consequently, the equality in Proposition 1 is false as a uniqueness statement; the correct claim is that P(A) is an RL-barycenter. The reader's single-valued-projection caveat is a different, degenerate failure; the non-uniqueness here is generic.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9827,"tokens_out":7324,"duration_ms":78583,"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":[{"comment":"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'.","section":"Section III, Definition 1 and Proposition 1"},{"comment":"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.","section":"Section III, Proposition 1"},{"comment":"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].","section":"Section IV"}],"minor_comments":[{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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.","section":"Section III, after Proposition 1"},{"comment":"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.","section":"Supplementary Materials B"},{"comment":"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.","section":"References / reproducibility"},{"comment":"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.","section":"Section II, notation"}],"recommendation":"major_revision","confidential_remarks":"The uniqueness issue in Proposition 1 is the main reason for major revision. It is a statement-level error that can be fixed by weakening the claim, but it affects the abstract, the introduction, and the interpretation of the experiments. The numerical comparison would also benefit from implementation details. If the authors are willing to reframe the contribution as 'the projected mean is an RL-barycenter' and add the necessary non-degeneracy assumptions, the paper is likely suitable for publication in a signal-processing or machine-learning venue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the core observation is correct and useful: with the projection retraction and the tangent projection as lifting, the arithmetic mean projected onto the manifold is a fixed point of the RL-barycenter equation. That gives a closed-form alternative to iterative R-barycenters, and the numerical experiments support it. Second, the paper says \"the\" RL-barycenter, but the fixed-point equation is not unique. The proof only shows existence. On Gr_{3,1} with 16 copies of e1e1^T, 3 of e2e2^T, 1 of e3e3^T, both e1e1^T and e3e3^T have zero mean lifting, so both are RL-barycenters. Same on St_{2,1} with e1 and e2: both ±(e1+e2)/√2. This is not a degenerate edge case; it is generic when the mean lifting can vanish at multiple points.\n\nThe paper does well: the RL-barycenter formalism is a clean generalization of R-barycenters, decoupling retraction from inverse. Proposition 1 is a short and correct proof of existence. The Grassmann projection result (top-k eigenvectors) is standard but the paper gives a self-contained proof in the supplement. The QR extension in the supplement is plausible but rests on a manual differential calculation; I did not verify every line, but it is likely correct. The numerical comparison is honest: on Stiefel their method beats polar/QR R-barycenters at larger noise, and on Grassmann it is nearly as good as the Riemannian mean. Code is provided.\n\nSoft spots, in order. (1) The uniqueness wording. Definition 1 defines an RL-barycenter as any solution; Proposition 1 calls the projected mean \"the\" barycenter. That is false. The correct statement is that the projected mean is an RL-barycenter. The paper's own fixed-point equation in Definition 1 already hints at this: any G with zero mean lifting qualifies. (2) The projection P must be single-valued for the arithmetic mean to be well-defined: on Stiefel the mean matrix needs full column rank, and on Grassmann the k-th and (k+1)-th eigenvalues must be distinct. The paper does not state these conditions. They are more likely to fail in small samples. (3) The numerical experiments are a bit light on implementation details: tolerances and iteration counts for the iterative R-barycenter solvers are not reported, but this is a minor issue for a letter.\n\nWho is it for: people in signal processing and ML who average data on Stiefel/Grassmann and currently run iterative fixed-point algorithms. They will get a cheaper estimator that is competitive in practice.\n\nRecommendation: send it to review. The core result is correct, the framework is a genuine contribution, and the uniqueness issue is a presentational fix, not a load-bearing flaw. A good referee will ask the authors to soften Proposition 1 to an existence claim and add the non-uniqueness discussion.","headline":"The projected mean is a real RL-barycenter, but it is not the only one; Proposition 1's uniqueness claim is too strong.","tokens_in":10339,"tokens_out":3223,"would_cite":true,"duration_ms":30282,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62R30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["RL-barycenter","Stiefel manifold","Grassmann manifold","projected arithmetic mean","retraction","lifting","polar decomposition","eigenprojector"],"falsifier":"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.","tokens_in":9329,"feed_emoji":"📐","tokens_out":13561,"duration_ms":126604,"temperature":0.7,"pith_summary":"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.","feed_headline":"Averaging on Stiefel and Grassmann reduces to one projection","feed_subtitle":"The new RL-barycenter replaces iterative manifold means with a single matrix projection at lower cost.","key_machinery":"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.","core_discovery":"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 $d\\,\\mathrm{qf}(G)[M-G]$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies projection-like retractions on matrix manifolds, including the identity $P_G(Z)=dP(G)[Z]$ that connects the projection retraction to the tangent-space lifting used in Proposition 1.","marker":"[19]"},{"why":"Defines R-barycenters and the Stiefel retractions (polar, QR, orthographic) whose inverse-retraction loop the closed-form projected mean replaces.","marker":"[17]"},{"why":"Provides the theorem that the Euclidean projection onto the Stiefel manifold is the orthogonal factor of the polar decomposition of the input matrix.","marker":"[20]"},{"why":"Supplies the retraction definition, the differential of the QR map, and the Stiefel tangent-space formulas used in the QR barycenter proof.","marker":"[9]"},{"why":"Supplies the Stiefel and Grassmann geometry, tangent spaces, and exponential and logarithm operators that frame the Fréchet mean and the projection-based averages.","marker":"[8]"},{"why":"Supplies the Grassmann tangent-space projection and the Riemannian distance used in the Grassmann experiments and in the supplementary proof of the projection map.","marker":"[13]"},{"why":"Introduces R-barycenters on the Grassmann manifold through tangent-bundle maps, giving the baseline against which the projected Grassmann mean is compared.","marker":"[18]"}],"fun_headline_variants":["RL-barycenters: arithmetic mean then project onto manifold","Manifold means simplified: one projection replaces iterations","Project the average: new barycenters for Stiefel and Grassmann","One projection does it: RL-barycenters for Stiefel & Grassmann"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["RL-barycenters: arithmetic mean then project onto manifold","Manifold means simplified: one projection replaces iterations","Project the average: new barycenters for Stiefel and Grassmann","One projection does it: RL-barycenters for Stiefel & Grassmann"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000799,"raw_usage":{"total_tokens":3506,"prompt_tokens":929,"completion_tokens":2577,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":2500}},"tokens_in":545,"tokens_out":2577,"duration_ms":18619,"temperature":1.0,"reasoning_tokens":2500,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:07:55.859938+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Absil and J","cited_arxiv_id":null,"evidence_quote":"Supplies projection-like retractions on matrix manifolds, including the identity $P_G(Z)=dP(G)[Z]$ that connects the projection retraction to the tangent-space lifting used in Proposition 1."},{"cited_title":"Kaneko, S","cited_arxiv_id":null,"evidence_quote":"Defines R-barycenters and the Stiefel retractions (polar, QR, orthographic) whose inverse-retraction loop the closed-form projected mean replaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the theorem that the Euclidean projection onto the Stiefel manifold is the orthogonal factor of the polar decomposition of the input matrix."},{"cited_title":"Edelman, T","cited_arxiv_id":null,"evidence_quote":"Supplies the Stiefel and Grassmann geometry, tangent spaces, and exponential and logarithm operators that frame the Fréchet mean and the projection-based averages."},{"cited_title":"Bendokat, R","cited_arxiv_id":null,"evidence_quote":"Supplies the Grassmann tangent-space projection and the Riemannian distance used in the Grassmann experiments and in the supplementary proof of the projection map."},{"cited_title":"Fiori, T","cited_arxiv_id":null,"evidence_quote":"Introduces R-barycenters on the Grassmann manifold through tangent-bundle maps, giving the baseline against which the projected Grassmann mean is compared."}],"review_version":1}