{"id":"cb7ff7bb-bce7-4dc1-968b-a61f1e41beae","arxiv_id":"1908.04233","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"On spheres of dimension at least two, Fréchet means can converge at rate n^{-1/6} even when data avoid the antipodal point, a 'geometrical smeariness' that is absent on the circle.","lead":"Averaging data on a sphere can converge much more slowly than usual, at the one-sixth root of the sample size instead of the square root. This paper shows this slowdown is caused by the sphere's curvature, not by data near the opposite point, and can even occur on spaces shaped like flat Euclidean space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 asserts an n^{-1/6} rate that the proof never establishes: the smeary CLT assumptions are not verified for the atomic-plus-hole example, and the stated theorem omits the tuning condition α=αβ.","rationale":"I read the paper as a constructive existence result: geometrical smeariness can occur even when the support avoids the cut locus. The strongest mathematical core—the derivative computations of the Fréchet function, the vanishing-Hessian construction, and the dimension-dependent hole bounds—is largely sound and matches the cited EH19 calculations. The most load-bearing weakness is not the uncomputed Lipschitz constants in Lemma A.8: finiteness of those constants is plausible from the displayed bounds and, if finite, yields a positive (if small) β0. The more serious gap is that the theorem's headline claim is an asymptotic rate, n^{-1/6}, and the paper provides no argument connecting the population Fréchet function's flatness to the sample Fréchet mean's scaling for this particular distribution. The paper itself flags that the CLT needs stricter assumptions than Definition 2.8, so this is not a stylistic omission. The theorem statement also misstates the quantifier over α; the proof uses α=αβ. These are fixable, so conditional accept remains the right verdict, but they are exactly the conditions that must be met before the central claim is established.","tokens_in":24283,"tokens_out":14812,"duration_ms":149730,"concrete_test":"Derive the sample-mean asymptotics for the exact distribution in Theorem 3.1 with m=5, β=β0/2, α=αβ by checking each condition of Eltzner-Huckemann (2019), Assumption 2.6 and Theorem 2.7: (i) the map x↦E[d²(exp_µ(x),X)] is four times differentiable at 0 with F''(0)=0<F''''(0); (ii) the class of functions {d²(exp_µ(x),·): ||x||<δ} is Donsker for the atomic-plus-annulus measure, and the envelope is square-integrable. If (ii) fails because of the atom or the sharp annulus boundary, then n^{-1/6} does not follow. If both hold, state the verification in an appendix and rerun the theorem with α=αβ made explicit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the existence of a unique 2-smeary Fréchet mean with asymptotic rate n^{-1/6} for the uniform-on-annulus-plus-atom distribution (Theorem 3.1). The proof establishes only local minimality (vanishing Hessian, positive fourth derivative) and a sketchy uniqueness argument via Lemma A.8. It never derives the n^{-1/6} scaling. The paper's Definition 2.8 notes that the smeary asymptotic CLT requires the stricter Assumption 2.6 of Eltzner and Huckemann (2019); Theorem 3.1 does not verify this assumption for a distribution with an atom at the mean and an indicator-supported annulus. Definition 2.8 alone gives a statement about the population Fréchet function, not about the sample-mean distribution. Moreover, Theorem 3.1 as written quantifies over arbitrary 0<α<1, but the proof requires the special choice α=αβ for each β; for arbitrary α the Hessian need not vanish and the claimed n^{-1/6} rate fails. These omissions leave the theorem's headline rate unproven, even though the underlying geometric construction may be sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a distinction between cut-locus smeariness and geometrical smeariness for Fréchet means on Riemannian manifolds. It claims that on the circle smeariness is cut-locus type, while on spheres S^m with m≥2 it is geometrical (Theorem 2.10). The main construction (Theorem 3.1) considers a random variable on S^m, m≥5, uniformly distributed on a spherical annulus L_{m,β} with total mass α plus a point mass at the north pole, and claims that for sufficiently small hole radius β the north pole is the unique Fréchet mean and is 2-smeary with asymptotic rate n^{-1/6}. The paper also proves a curse-of-dimensionality result for the admissible hole radius (Theorem 3.3), discusses finite sample smeariness, presents simulations on Kendall pre-shape space, and analyzes geomagnetic pole reversal data.","tokens_in":24478,"tokens_out":9741,"duration_ms":101214,"significance":"The geometrical-smeariness concept is original and, if established, would be an important contribution to non-Euclidean statistics: it shows that slow n^{-1/6} convergence can occur without probability mass near the cut locus and even on manifolds diffeomorphic to R^m. The supplement contains careful, largely self-contained integral computations for the second and fourth derivatives of the Fréchet function, and the paper is honest about the places where numerical bounds or conjectures are used. The simulations and the geomagnetic data analysis provide a useful practical illustration. At present, however, the formal definition of smeariness is not adequate as written, and the n^{-1/6} claim in Theorem 3.1 is not actually derived; these issues require substantial revision rather than minor polishing.","major_comments":[{"comment":"The definition of smeariness only imposes a lower bound: sup_{||x||<δ} |F(x)-F(0)| ≥ C_X δ^κ with κ>2. This condition is automatically satisfied by every Fréchet mean with positive-definite Hessian, because such an F satisfies F(x)-F(0) ≥ c||x||² in a neighborhood, so the supremum over a δ-ball is at least cδ², and for small δ<1 one has cδ² ≥ Cδ^κ for a suitable choice of C. Thus every nondegenerate mean would be classified as smeary, and the definition does not isolate the intended slower-than-n^{-1/2} phenomenon. The definition needs a two-sided growth condition, or an explicit requirement that all derivatives below order κ vanish at 0, to be mathematically meaningful.","section":"Definition 2.8"},{"comment":"Theorem 3.1 quantifies over arbitrary total mass 0<α<1 of the uniform annulus component, but the proof uses only the special value α=α_β defined by the condition ∂²F/∂ψ²(α_β,β,0)=0. For generic α the Hessian at the north pole need not vanish, and the claimed n^{-1/6} rate cannot hold; indeed for α<α_β one would expect standard n^{-1/2} asymptotics. The theorem should be restated with α=α_β, or it should be made explicit that the assertion applies only to the tuned value of α.","section":"Theorem 3.1, statement"},{"comment":"The asymptotic rate n^{-1/6} is asserted but never derived. Definition 2.8 concerns the population Fréchet function; the distribution of the sample Fréchet mean requires a smeary central limit theorem. The paper's own note after Definition 2.8 says that the smeary asymptotic theory relies on the stricter (Eltzner and Huckemann, 2019, Assumption 2.6), but Theorem 3.1 does not verify that assumption for a distribution with an atom at the mean and an indicator-supported annulus with a hole. In fact, Assumption 2.5(ii) requires a density in a neighborhood of Cut(μ), and the support L_{m,β} with β>0 explicitly excludes such a neighborhood. The theorem should either prove the needed CLT for this case or weaken the claim to '2-smeary in the sense of Definition 2.8' without the n^{-1/6} conclusion.","section":"Theorem 3.1, proof / Definition 2.8 note"},{"comment":"Theorem 2.12 states that for every ρ≥0 one can construct a random variable with density value ρ at the south pole and a non-smeary mean at the north pole. The proof, however, sets α=sinδ/(4π) and shows only that the south-pole density can be made arbitrarily large as δ→0. This establishes arbitrarily large density values, not that every nonnegative ρ is attained. A continuity or monotonicity argument over the full range of δ, or a modified construction covering small ρ, is needed; otherwise the theorem should be weakened to 'arbitrarily large ρ'.","section":"Supplement A.2.1 / Theorem 2.12"}],"minor_comments":[{"comment":"The sentence 'This is established by Lemmas A.3 and A.5 in the supplement' appears to cite the wrong lemmas; the local-minimum calculation for the annulus model with a hole is carried out in Lemmas A.6 and A.7.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The term '2-smeary' is used in Theorem 3.1 without being defined. Please define k-smeary, or explain the relation between the order κ in Definition 2.8 and the label '2-smeary' and the rate n^{-1/6}.","section":"Theorem 3.1 / Definition 2.8"},{"comment":"The heading 'A.2.1 Proof of Theorem 2.14' should refer to Theorem 2.12, not Theorem 2.14.","section":"Supplement A.2"},{"comment":"The sentence beginning 'No note that due to convexity...' appears to contain a typo ('No note' should likely be 'Note').","section":"Supplement A.2.1"},{"comment":"The Lipschitz constants L2, L3, L4 in Lemma A.8 are asserted to exist but are not explicitly bounded, and the threshold β0 in the proof of Theorem 3.1 depends on them. Since the proof argues finiteness, this is not fatal, but an explicit bound or a more transparent finiteness argument would make the uniqueness window quantitative.","section":"Lemma A.8 / Eq. (1)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper's main conceptual contribution is real: it separates cut-locus smeariness from geometrical smeariness, shows that on spheres S^m with m≥2 only the latter occurs, and constructs an example with a hole around the antipode whose Fréchet function has vanishing Hessian and positive fourth derivative. That is genuinely new and the derivative computations are careful. The second thing is less good: Theorem 3.1, as stated, asserts an n^{-1/6} sample rate for the atomic-plus-annulus variable, but the proof does not establish any sample rate. It only shows the population Fréchet function has a local minimum at the north pole after one tunes α to α_β, and the theorem statement omits that tuning condition. For arbitrary α the Hessian need not vanish, so the claimed rate fails. To get n^{-1/6} you need the smeary CLT from Eltzner and Huckemann (2019) to apply to this particular distribution, and that is never verified. The stress-test note lands on this correctly.\n\nWhat the paper does well: the distinction itself is worth having, Theorem 2.10 is simple and convincing, and the corollary that smeariness can occur on a manifold diffeomorphic to Euclidean space is surprising. The curse-of-dimensionality result in Theorem 3.3 is also interesting. The writing is clear and the literature is engaged honestly, including conjectures and caveats.\n\nSoft spots, in proportion: the uniqueness proof relies on Lipschitz constants that are only shown to exist, not evaluated, so the explicit threshold β0 is a black box. That is acceptable for an existence argument, but it is a gap between what is proven and what is claimed. The simulation omits the tuning parameter α, which limits reproduction. The real-data section is finite-sample bootstrap evidence only, and the paper is appropriately cautious about that.\n\nThis paper is for statisticians working on Fréchet means, shape analysis, and directional data. The central construction is plausible and the conceptual framework is useful, but the headline theorem needs revision: either add the α=α_β condition and prove or explicitly cite the smeary CLT for this example, or rephrase the claim as a conjecture with the local-minimum result stated separately. I would send this to a serious referee rather than desk-reject.","headline":"Useful conceptual distinction and a clever construction, but Theorem 3.1 overclaims the n^{-1/6} rate the proof never establishes.","tokens_in":25038,"tokens_out":3608,"would_cite":true,"duration_ms":38237,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62R30","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"On spheres of dimension five and higher, a Fréchet mean can converge at $n^{-1/6}$ purely because of curvature, even when data avoids the cut locus.","keywords":["Fréchet mean","geometrical smeariness","cut locus smeariness","central limit theorem on manifolds","spheres","curvature","n^{-1/6} rate","finite sample smeariness"],"falsifier":"For $m=5$, numerically evaluate the exact Fréchet function $F(\\alpha_\\beta,\\beta,\\psi)$ for the uniform annulus plus point mass at the north pole, choosing $\\alpha_\\beta$ so the Hessian vanishes at $\\psi=0$, and check whether $\\partial^2 F/\\partial\\psi^2 > 0$ for all $\\psi\\in(0,\\pi]$ when $\\beta\\le\\beta_0$. If the maximal allowed hole radius is not strictly positive, or if simulations of the sample mean do not show variance scaling like $n^{-1/3}$, the uniqueness-and-rate claim of Theorem 3.1 would be refuted.","tokens_in":24023,"feed_emoji":"🌐","tokens_out":16105,"duration_ms":142383,"temperature":0.7,"pith_summary":"The paper establishes a new kind of failure of the central limit theorem on curved data spaces. On the circle, the anomalously slow $n^{-1/6}$ fluctuation of a Fréchet mean (2-smeariness) requires a special value of the probability density at the antipodal point, the cut locus. The paper proves that on spheres of dimension $m\\ge 5$, the same slow rate can occur for purely geometric reasons: the mean sits at the north pole, while the data is uniform on the southern hemisphere with a hole of radius $\\beta$ around the south pole plus a point mass at the north pole. For any sufficiently small $\\beta$ the north pole is the unique Fréchet mean, and its asymptotic fluctuation scale is $n^{-1/6}$. This shows smeariness is not an artifact of mass sitting at the cut locus, and it can even be transplanted to a manifold diffeomorphic to Euclidean space.","feed_headline":"Sphere means can converge at $n^{-1/6}$","feed_subtitle":"Curvature alone can make Fréchet means on high-dimensional spheres converge at the slow $n^{-1/6}$ rate.","key_machinery":"The machinery is the rotationally symmetric Fréchet function $F(\\alpha,\\beta,\\psi)$, where $\\psi$ is the polar angle to the north pole $\\mu$. Because the measure is invariant under rotations about the polar axis, proving smeariness reduces to showing $\\partial^2 F/\\partial\\psi^2$ vanishes at $\\psi=0$ while $\\partial^4 F/\\partial\\psi^4>0$, so the Fréchet function grows like $\\psi^4$ and the sample mean fluctuates at $n^{-1/6}$. The spherical annulus $L_{m,\\beta}$ is the key object: its hole around the south pole removes the cut locus from the support, isolating the curvature contribution. Geometrical smeariness is smeariness in which the negative Hessian contribution from the cut locus is absent: either $F=G$ in a neighborhood of the mean or $\\mathrm{Hess}(F-G)(0)\\ge 0$. The proof that the local minimum is global uses Lipschitz bounds (Lemma A.8) on how the second, third, and fourth derivatives of $F$ change with $\\beta$, yielding the threshold $\\beta_0 = \\min(c_m/(2L_4), (1/L_2)\\,\\partial^2 F/\\partial\\psi^2(\\alpha_0,0,\\pi/3))$.","core_discovery":"On its own terms, the paper's central discovery is that smeariness on spheres is geometrical, not cut-locus. Theorem 2.10 states that the circle $S^1$ only admits cut locus smeariness, while every sphere $S^m$ with $m\\ge 2$ only admits geometrical smeariness: near the mean, the Fréchet function using only geodesics that avoid the cut locus coincides with the full Fréchet function, because geodesics can circumvent the antipodal point. Theorem 3.1 then constructs, for $m\\ge 5$, a random variable with a unique 2-smeary Fréchet mean at the north pole whose support excludes a neighborhood of the antipodal point: uniform mass $\\alpha$ on the spherical annulus $L_{m,\\beta} = \\{q : \\arccos\\langle q,\\mu\\rangle\\in[\\pi/2,\\pi-\\beta]\\}$ plus point mass $1-\\alpha$ at $\\mu$, with $\\alpha$ tuned so the Hessian of the Fréchet function vanishes at $\\mu$ while its fourth derivative is positive. The resulting asymptotic rate is $n^{-1/6}$, and a corollary deforms the opposite hemisphere to build the same phenomenon on a manifold diffeomorphic to $\\mathbb{R}^m$. A further theorem gives a curse of dimensionality: as $m$ grows, the hole radius may approach $\\pi/2$, so in high dimension even support barely exceeding a hemisphere can produce smeary local means.","pith_inferences":["Because the Lipschitz constants in the proof are not evaluated, the practical size of the hole radius $\\beta_0$ in concrete dimensions is unknown; computing them numerically would show whether the $n^{-1/6}$ regime is robust or confined to a tiny parameter window.","The same vanishing-Hessian plus positive-quartic mechanism should produce geometrically smeary means on other positively curved, rotationally symmetric spaces, beyond the spheres treated here.","For practitioners, checking the curvature of the empirical Fréchet function at the estimated mean may be a more useful diagnostic for smeariness than inspecting the density near the antipodal point."],"forward_implications":["On $S^m$ with $m\\ge 5$, smeariness can occur even when the support of the random variable has a hole around the cut locus, so the antipodal density is irrelevant.","A Fréchet mean with vanishing Hessian and positive quartic term has $n^{-1/6}$ fluctuations; the paper shows such means exist concretely on spheres and on manifolds diffeomorphic to $\\mathbb{R}^m$.","Deforming the opposite hemisphere to a flat space yields smeary means on a manifold diffeomorphic to $\\mathbb{R}^m$, so topological triviality does not prevent the phenomenon.","In high dimension, the allowed hole radius grows toward $\\pi/2$: random variables whose support barely exceeds a hemisphere can already have smeary local means.","Finite-sample smeariness of large magnitude is possible for moderately spread spherical data, and the bootstrap analysis of geomagnetic pole-reversal data finds 17 of 151 data sets consistent with it."],"supporting_citations":[{"why":"Supplies the smeary central limit theorem and the formula $c_m = \\alpha V_{m+1}/V_m\\,(m-1)/(m+2)$ used to get a positive fourth derivative at the mean.","marker":"Eltzner and Huckemann (2019)"},{"why":"Introduced smeariness for circle-valued data and computed the cut-locus Hessian contribution that Theorem 2.10 contrasts with the spherical case.","marker":"Hotz and Huckemann (2015)"},{"why":"Theorem 5.52 provides the M-estimator rate framework behind Definition 2.8's notion of smeariness as a non-standard $n^{-\\tau}$ rate.","marker":"van der Vaart (2000)"},{"why":"Defines finite sample smeariness and provides the bootstrap variance-scaling diagnostic used on the geomagnetic pole data.","marker":"Hundrieser et al. (2020)"},{"why":"Shows the cut locus of a Fréchet mean carries no probability mass, used by the paper to interpret the real-data findings.","marker":"Le and Barden (2014)"},{"why":"Establishes that positively curved spaces generically have $n\\mathrm{Var}[\\hat\\mu_n]>\\mathrm{Var}[X]$, giving the finite-sample smeariness context for Theorem 4.3.","marker":"Pennec (2019)"}],"fun_headline_variants":["Curvature alone slows Frechet means on spheres to n^{-1/6}","Geometrical smeariness: curvature gives n^{-1/6} on spheres","Why Frechet means on spheres slow to n^{-1/6}","Curvature, not antipodal density, sets sphere smeariness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the local smeary minimum is the unique global mean rests on Lipschitz bounds that are asserted to exist but never explicitly evaluated, and it also assumes the smeary CLT conditions apply to a distribution with a point mass and a hole.","fun_headline_variants_meta":{"raw":{"variants":["Curvature alone slows Frechet means on spheres to n^{-1/6}","Geometrical smeariness: curvature gives n^{-1/6} on spheres","Why Frechet means on spheres slow to n^{-1/6}","Curvature, not antipodal density, sets sphere smeariness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001218,"raw_usage":{"total_tokens":5124,"prompt_tokens":1175,"completion_tokens":3949,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":791,"completion_tokens_details":{"reasoning_tokens":3866}},"tokens_in":791,"tokens_out":3949,"duration_ms":28883,"temperature":1.0,"reasoning_tokens":3866,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:48:01.028994+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $m=5$, numerically evaluate the exact Fréchet function $F(\\alpha_\\beta,\\beta,\\psi)$ for the uniform annulus plus point mass at the north pole, choosing $\\alpha_\\beta$ so the Hessian vanishes at $\\psi=0$, and check whether $\\partial^2 F/\\partial\\psi^2 > 0$ for all $\\psi\\in(0,\\pi]$ when $\\beta\\le\\beta_0$. If the maximal allowed hole radius is not strictly positive, or if simulations of the sample mean do not show variance scaling like $n^{-1/3}$, the uniqueness-and-rate claim of Theorem 3.1 would be refuted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced smeariness for circle-valued data and computed the cut-locus Hessian contribution that Theorem 2.10 contrasts with the spherical case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Theorem 5.52 provides the M-estimator rate framework behind Definition 2.8's notion of smeariness as a non-standard $n^{-\\tau}$ rate."},{"cited_title":"Finite Sample Smeariness of Fr\\'echet Means and Application to Climate","cited_arxiv_id":"2005.02321","evidence_quote":"Defines finite sample smeariness and provides the bootstrap variance-scaling diagnostic used on the geomagnetic pole data."}],"review_version":1}