{"id":"4dbcdaff-f6f2-4332-9557-f4aef58db08d","arxiv_id":"2602.19923","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The polar-light retraction maps tangent vectors to the Stiefel manifold via U expm(U^T ξ) + (I-UU^T)ξ normalized, giving a second-order retraction with an explicit inverse via SVD and matrix logarithm.","lead":"A new 'polar-light' retraction on the Stiefel manifold combines second-order accuracy under the Euclidean metric with a closed-form inverse, where the inverse costs about the same as the forward map. It may speed up manifold interpolation and averaging, where the logarithm map is used repeatedly.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'first closed-form inverse under Euclidean metric' claim is under-supported and internally contradicted by the manuscript's own text.","rationale":"The reader's verdict was CONDITIONAL, and I agree that conditional acceptance is appropriate. However, the single most load-bearing concern is not the uncharacterized domain of the inverse chart (the reader's weakest_assumption), but the unsupported, self-contradictory novelty claim. The paper's Abstract and Introduction directly disagree about whether the Cayley retraction has a closed-form inverse. Since the central contribution is precisely that no prior retraction offers Euclidean-second-order accuracy plus a closed-form inverse, the paper must present a reliable survey. The domain issue is real but standard for a retraction (local diffeomorphism); the 'first' claim is what distinguishes the paper. A targeted literature check and re-derivation of the Cayley retraction's Euclidean order would settle the matter. The numerical experiments are preliminary but do not resolve the novelty question. Thus, the verdict stays CONDITIONAL: the mathematics appears correct, but the authors must correct the internal contradiction and substantiate the 'first' assertion before acceptance.","tokens_in":8689,"tokens_out":19607,"duration_ms":166297,"concrete_test":"Perform an analytical check: derive the inverse of the Cayley retraction from [15] and compute its Taylor expansion under the Euclidean metric (β=1) using eq. (3.8). If the Cayley retraction is second-order under β=1 and has a closed-form inverse, then the paper's 'first' claim is false. If it is only first-order under Euclidean (or its inverse is not closed-form), then the new retraction's niche is preserved; either way, the contradictory statements about the Cayley inverse in the Abstract and §1 must be resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the polar-light retraction is the first Stiefel retraction to combine second-order accuracy under the Euclidean metric with a closed-form inverse. This novelty claim is weakened by an explicit internal contradiction: the Abstract states that the Cayley retraction 'is second-order accurate under the canonical metric and features a closed-form inverse,' while the Introduction (lines 30–35) says 'for the ... Cayley retraction, we are not aware of any published work on computing the inverse' and asserts that quasi-geodesics are 'the only ones on the list that feature a closed form inverse.' Both cannot be true. More importantly, the paper does not systematically rule out existing retractions (e.g., variants of the Cayley retraction) that might already be second-order under the Euclidean metric and have a closed-form inverse. If such a retraction exists, the 'first' claim collapses, leaving only an incremental contribution. The mathematical construction itself appears sound, but the historical/literature claim that anchors the paper's significance is not adequately supported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new retraction on the compact Stiefel manifold, called the polar-light retraction. For a base point Û and tangent vector ξ, the forward map is φ_Û(ξ) = (Û expm(Ûᵀξ) + (I − ÛÛᵀ)ξ)(I + ξᵀ(I − ÛÛᵀ)ξ)^{-1/2}. The authors prove that this map is a second-order retraction under the Euclidean metric, and they give an explicit inverse formula ψ_Û(U) based on a singular value decomposition and a matrix logarithm. Numerical experiments compare the retraction and its inverse with the classical polar factor retraction in terms of accuracy and runtime. The paper's central claim is that this is the first Stiefel retraction that is simultaneously second-order accurate under the Euclidean metric and has a closed-form inverse.","tokens_in":8925,"tokens_out":11574,"duration_ms":103025,"significance":"The mathematical development is largely sound and transparent: the Taylor expansion in Lemma 3.3 correctly matches the Riemannian exponential second derivative for the Euclidean metric, and the inverse identities in Lemma 3.2 are algebraically verified. Providing reproducible Python code is a clear strength. If the novelty claim is properly substantiated and the domain of the inverse is characterized, the polar-light retraction would be a useful tool for manifold interpolation, barycenter computation, and other applications requiring an explicit inverse retraction. However, the significance depends critically on two issues: the paper's own text contradicts itself about the Cayley retraction's inverse, and the inverse formula's domain of validity is left unspecified.","major_comments":[{"comment":"The abstract (as printed at the top) states that the Cayley retraction 'is second-order accurate under the canonical metric and features a closed-form inverse,' while §1 (lines 30–35) states that for the Cayley retraction 'we are not aware of any published work on computing the inverse' and that quasi-geodesics 'are the only ones on the list that feature a closed form inverse.' These statements are mutually contradictory. Because the paper's main novelty claim is being 'the first' with these properties under the Euclidean metric, this contradiction is load-bearing. The authors must correct the abstract, clarify whether the novelty is relative to the Euclidean metric only, and provide a more systematic literature check on existing retractions with closed-form inverses.","section":"Abstract and §1"},{"comment":"The inverse formula is not well-defined on the entire Stiefel manifold. It requires ÛᵀU to have full rank, otherwise the SVD factor S is singular and S^{-1} is undefined. Additionally, the matrix M Rᵀ = ÛᵀU(UᵀÛÛᵀU)^{-1/2} is only in O(p), not necessarily SO(p), and the principal matrix logarithm is real only when this matrix has no negative real eigenvalues. The paper merely states that B is 'a relative open, path-connected neighborhood' without characterizing it. The experiments use endpoints at distance π/2, where ÛᵀU can be singular in low-dimensional examples. The authors must specify conditions under which (3.5)–(3.7) produce a real tangent vector and discuss branch selection for the logarithm.","section":"Lemma 3.1 and Eqs. (3.5)–(3.7)"},{"comment":"The argument that det(R)=det(U₁(U₁ᵀU₁)^{-1/2})=+1 uses a continuous path from E to U inside B. However, if the path passes through a point where U₁ is singular, the polar factor R is not defined. B must be chosen inside the open set {U : det(U₁)≠0}, and this restriction should be stated explicitly. The current text gives the impression that B can be any small neighborhood, but the chart construction requires avoiding both singular U₁ and orthogonal factors with negative real eigenvalues.","section":"Lemma 3.1, proof of det(R)=+1"}],"minor_comments":[{"comment":"The title contains a grammatical error: 'An new' should be 'A new'.","section":"Title"},{"comment":"Typo: 'paramaterization' should be 'parametrization' or 'parameterization'.","section":"Lemma 3.2"},{"comment":"Typo: 'Forbenius' should be 'Frobenius'.","section":"Section 4"},{"comment":"Reference [9] lists the same URL twice; this should be cleaned up.","section":"References"},{"comment":"The numerical experiments are presented as single runs without error bars or repeated trials. Since the claims about accuracy are empirical, a few repetitions or confidence intervals would strengthen the presentation.","section":"Section 4"},{"comment":"The term 'closed-form' is used for formulas involving SVD and matrix logarithm, which are iterative in floating-point arithmetic. The authors should clarify that 'closed-form' means 'no iterative solve of a matrix equation' in this context.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The central mathematical construction is sound and publishable in principle. The main obstacles are the internal contradiction about the Cayley retraction's inverse, which undermines the novelty claim, and the uncharacterized domain of the inverse formula. Both are fixable in revision. I would encourage the editor to send the paper back with these two points as required conditions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jensen and Zimmermann give a new retraction on the Stiefel manifold, the polar-light retraction, built by replacing the upper p×p block in the classical polar factor retraction with expm(A). The clean part is the inverse: an explicit SVD-based formula that is genuinely closed-form, and the Taylor expansion in Lemma 3.3 shows second-order accuracy under the Euclidean metric. The code is on GitHub, so the numerical comparison with the polar factor retraction is reproducible. That part is solid and useful.\n\nThe soft spot is the novelty claim. The abstract says the Cayley retraction 'features a closed-form inverse,' while the introduction says the authors are 'not aware of any published work on computing the inverse' for Cayley and that quasi-geodesics are the only ones with a closed-form inverse. Both cannot be true. This contradiction sits directly on top of the paper's central assertion that the new retraction is the first with second-order accuracy under the Euclidean metric and a closed-form inverse. The authors do not systematically survey the retraction literature to rule out existing examples; they assert it. That needs to be either proven or softened.\n\nA second soft spot, smaller, is the domain of the inverse chart. The logm in (3.7) requires the orthogonal factor M R^T to have no negative real eigenvalues. The paper defines B as a 'suitable neighborhood' but never characterizes it. For a local chart that's acceptable, but it limits the practical claim of an 'efficient logarithm' for arbitrary pairs (U0,U1). The numerical experiments only compare against the polar factor retraction, not against Cayley or quasi-geodesics, so the 'closer to geodesics' conclusion is narrower than the abstract implies.\n\nOverall: the mathematics is correct and the closed-form inverse is the real contribution. The paper is worth a serious referee, but the authors need to clean up the contradictory statements about the Cayley retraction and be more careful about what 'first' means. I'd send it to review with the expectation of a revision.","headline":"A sound, incremental retraction with a genuinely closed-form inverse, but the paper's own text contradicts its 'first' claim and the chart domain is vague.","tokens_in":9337,"tokens_out":3664,"would_cite":true,"duration_ms":33067,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A16","15B10","53Z50","65D05","65F60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The polar-light retraction gives the Stiefel manifold its first second-order retraction with a closed-form inverse, enabling explicit round-trip coordinates on the manifold of orthogonal frames.","keywords":["Stiefel manifold","retraction","polar decomposition","closed-form inverse","Euclidean metric","matrix logarithm","manifold interpolation","Riemannian computing"],"falsifier":"Take $p=1$ (the sphere) with $U_0 = e_1$ and $U_1 = -e_1$. Then $\\hat U^T U = -1$, and the inverse formula (3.7) calls logm on a scalar with no principal logarithm, so the closed-form inverse fails. More generally, any pair of Stiefel points whose Procrustes rotation $M R^T$ has a $-1$ eigenvalue breaks the inverse formula.","tokens_in":8609,"feed_emoji":"🔄","tokens_out":4752,"duration_ms":40289,"temperature":0.7,"texified_at":"2026-08-05T20:57:25.429769+00:00","pith_summary":"This paper introduces the polar-light retraction, a map from tangent vectors to the Stiefel manifold of orthogonal frames built from a polar decomposition of a small block. The central claim is that this retraction is the first that is second-order accurate under the Euclidean metric and also has a closed-form inverse. If true, it provides an explicit, non-iterative logarithm-style map for the Stiefel manifold, replacing the Sylvester or Lyapunov equation solvers previously needed for the polar factor retraction. Numerical experiments suggest the new retraction tracks Riemannian geodesics more closely than the classical polar factor retraction, especially in higher dimensions.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":4292,"prompt_tokens":721,"completion_tokens":3571,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":721,"completion_tokens_details":{"reasoning_tokens":2891}},"feed_headline":"Closed-form inverse found for a second-order Stiefel retraction","feed_subtitle":"Explicit round-trip maps on orthogonal frames replace iterative solvers for interpolation and averaging.","key_machinery":"The polar-light retraction: a map that exponentiates only the small p×p skew-symmetric block $A = \\hat U^T \\xi$ of a tangent vector, forms $\\hat U \\exp(A) + (I - \\hat U\\hat U^T)\\xi$, and normalizes with $\\left(I + \\xi^T(I - \\hat U\\hat U^T)\\xi\\right)^{-1/2}$ to land on the Stiefel manifold. The inverse uses the SVD $M S R^T = \\hat U^T U$ and the matrix logarithm of the orthogonal Procrustes rotation $M R^T$ to recover $A$ and $B$, making the inverse closed-form.","core_discovery":"The paper defines a coordinate chart centered at any Stiefel point $\\hat U$ via the forward map $\\phi_{\\hat U}(\\xi) = \\left(\\hat U \\exp(\\hat U^T \\xi) + (I - \\hat U\\hat U^T)\\xi\\right)\\left(I + \\xi^T(I - \\hat U\\hat U^T)\\xi\\right)^{-1/2}$ and its inverse $\\psi_{\\hat U}(U)$, which is computable in closed form from the SVD of $\\hat U^T U$. The authors claim that $\\phi$ is second-order accurate under the Euclidean metric ($\\beta=1$ in the metric family of the paper) and that $\\psi$ is the exact inverse on a neighborhood of the base point. This makes the pair the first Stiefel retraction with both second-order accuracy and a closed-form inverse, giving explicit local coordinates for the manifold.","pith_inferences":["The same small-block polar decomposition construction could likely be adapted to other reductive homogeneous spaces, giving explicit retraction/log pairs beyond the Stiefel manifold.","Because the inverse is explicit, tasks that repeatedly need the logarithm, such as intrinsic statistics or manifold regression, may become practical for larger p than before.","The paper's numerical results suggest the advantage over the polar factor retraction grows with p; a systematic study of the error versus p and distance would clarify when the extra forward cost is worthwhile.","Replacing expm/logm with Cayley approximations, as the authors suggest, would preserve second order only if the square-root factor is adjusted; the paper notes this but does not test it."],"forward_implications":["Computing the inverse retraction no longer requires solving a Sylvester or Lyapunov matrix equation; the cost is comparable to evaluating the retraction itself.","Riemannian barycenters and interpolation on the Stiefel manifold can be computed with explicit, non-iterative round-trip maps under the Euclidean metric.","The explicit inverse reveals how the skew-symmetric A-block and rectangular B-block enter the local coordinates, which may simplify error analysis in manifold optimization.","Because only p×p matrix functions appear, the method scales well when n >> p, with large matrices handled by simple multiplications."],"fun_headline_variants":["Stiefel retraction with closed-form inverse and second-order accuracy","New Stiefel retraction: SVD inverse makes round-trip exact","Closed-form inverse retraction on Stiefel gives exact round-trips","Stiefel retraction: SVD-based inverse is second-order accurate"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The closed-form inverse is only valid on the uncharacterized neighborhood where the Procrustes rotation between the two Stiefel points has no eigenvalue exactly -1, so the principal matrix logarithm is well-defined.","fun_headline_variants_meta":{"raw":{"variants":["Stiefel retraction with closed-form inverse and second-order accuracy","New Stiefel retraction: SVD inverse makes round-trip exact","Closed-form inverse retraction on Stiefel gives exact round-trips","Stiefel retraction: SVD-based inverse is second-order accurate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000833,"raw_usage":{"total_tokens":3463,"prompt_tokens":728,"completion_tokens":2735,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":2656}},"tokens_in":472,"tokens_out":2735,"duration_ms":16059,"temperature":1.0,"reasoning_tokens":2656,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T21:27:23.331222+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $p=1$ (the sphere) with $U_0 = e_1$ and $U_1 = -e_1$. Then $\\hat U^T U = -1$, and the inverse formula (3.7) calls logm on a scalar with no principal logarithm, so the closed-form inverse fails. More generally, any pair of Stiefel points whose Procrustes rotation $M R^T$ has a $-1$ eigenvalue breaks the inverse formula.","supporting_citations":[],"review_version":1}