{"id":"02088c79-3de1-40e7-b0f0-d1e5307e8a35","arxiv_id":"2411.14899","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The tangent-free rewrite of the Hauberg density is algebraically valid, but the identification of its contours with standard spherical ellipses is not.","lead":"The authors rewrite the anisotropic spherical normal density without tangent-plane coordinates and claim its contours are true spherical ellipses. The rewrite is clean, but the ellipse claim relies on treating the log map as a global isometry, which it is not.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4's conclusion that Eq. (7) defines spherical ellipses rests on the false claim that Log_mu is a global isometry; the tangent-plane ellipse does not pull back to a geodesic ellipse, so the central advertised result is unsupported and false.","rationale":"I read Section 3 first and confirmed that the algebraic passage from Hauberg's tangent-space density to the tangent-free form (2) is correct: the identity Log_mu(x)^T Lambda Log_mu(x) = arccos^2(x^T mu)[lambda1(x^T eta)^2+lambda2(x^T xi)^2]/[(x^T eta)^2+(x^T xi)^2] follows from the radial structure of the logarithm map. The serious problem is Section 4. The paper claims that because the image of E under Log_mu is a Euclidean ellipse, the inverse image E is a spherical ellipse. This is exactly the step that fails. Log_mu is not an isometry between the tangent plane and the sphere; it only preserves radial distances from mu. Consequently, a Euclidean ellipse in the tangent space does not pull back to a locus of constant sum of geodesic distances. I verified this by comparing the two polar equations at phi=pi/4. For finite alpha,beta the radial angles differ, e.g. alpha=pi/4, beta=pi/6 gives theta_E approx 0.6161 versus theta_S approx 0.6155. Thus Eq. (7) is not an exact equation for a spherical ellipse under the paper's own definition (3), and the density contours of (2) are not spherical ellipses. The algebraic reformulation in Eq. (2) remains useful, but the advertised geometric conclusion is false. The reader's verdict of REJECT with high confidence is correct; the reader's weakest assumption identifies the same invalid isometry step, although the more decisive check is the direct comparison of the radial equations rather than only the focal-distance discrepancy.","tokens_in":6481,"tokens_out":30932,"duration_ms":295263,"concrete_test":"For a fixed pair (alpha,beta), e.g. alpha=pi/4, beta=pi/6, evaluate Eq. (7) at azimuth phi=pi/4: it gives theta_E^2=2/(1/alpha^2+1/beta^2) approx 0.3796, so theta_E approx 0.6161. Independently derive the spherical ellipse from the constant-sum definition with foci at angular distance Gamma=arccos(cos alpha/cos beta): at phi=pi/4, sin^2 theta_S=2/(1/sin^2 alpha+1/sin^2 beta) approx 1/3, so theta_S approx 0.6155. Since theta_E != theta_S, Eq. (7) is not the spherical ellipse. The same analytic comparison settles the issue for any parameter values, e.g. alpha=1.5, beta=0.5 where the discrepancy is larger.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing step is the sentence after Eq. (7): 'Since Log_mu is an isometry, this implies...'. Log_mu preserves distances from mu only, not pairwise geodesic distances: dG(x,mu)=||Log_mu(x)||, but dG(x,f) != ||Log_mu(x)-Log_mu(f)|| for arbitrary f. The image of E under Log_mu is indeed the Euclidean ellipse (y·eta)^2/alpha^2+(y·xi)^2/beta^2=1 in T_mu, with Euclidean foci at ±sqrt(alpha^2-beta^2) eta. A true spherical ellipse with semimajor angle alpha and semiminor angle beta has foci at angular distance Gamma=arccos(cos alpha/cos beta), not sqrt(alpha^2-beta^2). For alpha=pi/4, beta=pi/6, these are about 0.6155 and 0.5854; for alpha=1, beta=0.6, about 0.859 and 0.800. The curves already differ at azimuth phi=pi/4: Eq. (7) gives theta=(0.5(1/alpha^2+1/beta^2))^{-1/2}, whereas the constant-sum spherical ellipse condition gives sin^2 theta=2/(1/sin^2 alpha+1/sin^2 beta). These are not equal, so E is not the locus dG(x,f1)+dG(x,f2)=2alpha for the claimed foci, nor any spherical ellipse with the stated axes. The tangent-free density (2) remains a valid algebraic reformulation, but the 'exactly spherical ellipses' claim is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a tangent-space-free expression for Hauberg's anisotropic spherical normal distribution, obtained by rewriting the Mahalanobis exponent of the log-map coordinates directly in terms of dot products on the sphere (Section 3, Eq. (2)). It then claims that the level sets of this density are exactly spherical ellipses, and proposes Eq. (7) as a new characterization of spherical ellipses (Section 4). The algebraic reformulation leading to Eq. (2) is correct, but the geometric conclusion is not. The argument that the exponential image of a Euclidean tangent-space ellipse is a spherical ellipse relies on treating the log map as a global isometry, which it is not; the paper's own focal-distance relation in Eq. (5) contradicts the Euclidean focal distance used in the derivation of Eq. (7).","tokens_in":6786,"tokens_out":13637,"duration_ms":120848,"significance":"If correct, the ellipse-contour claim would give the anisotropy parameters of the geodesic normal distribution a clear geometric interpretation and would provide a new equation for spherical ellipses. The tangent-free density formula (2) is indeed a valid and potentially convenient algebraic reformulation of Hauberg's density. However, the central geometric claim fails: Eq. (7) defines the exponential image of a Euclidean ellipse, which is not the constant-sum-of-geodesic-distances locus defined in Eq. (3). The paper's main advertised contribution, that the density contours are true spherical ellipses, is therefore unsupported and false.","major_comments":[{"comment":"The sentence 'Since Log_mu is an isometry' uses the log map in a way that is false. Log_mu preserves radial distances from mu, in the sense that dG(x,mu)=||Log_mu(x)||, but it does not preserve distances between two arbitrary points. The argument requires exactly such pairwise distance preservation in order to transfer the focal property of the Euclidean ellipse E* to a sum of geodesic distances on the sphere: one would need dG(x,Exp_mu(gamma eta))=||Log_mu(x)-gamma eta|| for points off the radial geodesic, which is not true on a curved sphere. Consequently, the set E defined by Eq. (7) is the exponential image of a Euclidean ellipse, but it is not the locus defined by Eq. (3). This invalidates the conclusion that the density contours of fGN in Eq. (2) are spherical ellipses.","section":"Section 4, after Eq. (7)"},{"comment":"The two characterizations are mutually inconsistent regarding the focal distance. In Eq. (5), a spherical ellipse with semiaxes alpha and beta has focus angle gamma satisfying cos(gamma)=cos(alpha)/cos(beta). In the tangent-space derivation leading to Eq. (7), the Euclidean ellipse E* has semifocal distance gamma_E=sqrt(alpha^2-beta^2). For alpha=pi/4 and beta=pi/6 these are approximately 0.6155 and 0.5854, respectively; they agree only in the circular limit beta=alpha. Since Log_mu is not an isometry, there is no mechanism by which the Euclidean focal distance becomes the correct geodesic focal distance. Thus Eq. (7) and Eq. (3) define different curves in general, so the claim that Eq. (7) represents a spherical ellipse with the stated axes is not correct.","section":"Section 4, Eqs. (5) and (7)"}],"minor_comments":[{"comment":"The word 'fuction' should be 'function'.","section":"Section 2"},{"comment":"Equation (6) is printed without visible fraction bars in the text; the intended equation appears to be x_1^2 cos^2(gamma)/cos^2(alpha) + x_2^2 sin^2(gamma)/sin^2(alpha) = 1. Please fix the typesetting.","section":"Section 4, Eq. (6)"},{"comment":"The symbol gamma is used both for the Euclidean semifocal distance in the tangent-space ellipse E* and for the geodesic focus angle in Eqs. (3)-(5). This notation collision obscures the inconsistency between the two uses; distinct symbols should be introduced.","section":"Section 4, after Eq. (7)"},{"comment":"The statement 'This map is an isometry' is imprecise. Log_mu is a radial isometry with respect to mu (it preserves distances from mu), but it is not a global isometry between the tangent space and the sphere. Clarifying this would help prevent the misuse in Section 4.","section":"Section 2"}],"recommendation":"reject","confidential_remarks":"The reader's assessment matches my reading of the manuscript. The algebraic reformulation in Eq. (2) is correct and could be a modest useful observation, but the paper's main claim—that the density contours are spherical ellipses—relies on a false global-isometry assumption and cannot be repaired by a local correction. A revision limited to the algebraic reformulation would be a very short note of limited scope. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the algebra in equations (1)-(2) is correct and gives a tangent-free expression for Hauberg's anisotropic geodesic normal. But the advertised new result, that the density contours are exactly spherical ellipses, is false. The argument in Section 4 relies on the claim that Log_mu is an isometry, and that is only true radially: it preserves distances from mu, not distances between arbitrary points.\n\nWhat the paper does well: it clearly derives the reformulation, and the discussion of spherical ellipses in equations (3)-(6) is a reasonable survey. If the only goal were to present Hauberg's density without tangent-space coordinates, this note would be fine. The authors are also honest that (2) is a reformulation of Hauberg's construction, not a new distribution.\n\nWhere it falls: Section 4 pulls a Euclidean ellipse back from T_mu and asserts that because Log_mu is an isometry, the pulled-back curve satisfies d_G(x,f1)+d_G(x,f2)=2alpha for f_i=Log_mu^{-1}(±gamma eta). That step is invalid. The log map does not preserve pairwise distances, so the image of a Euclidean ellipse is not generally a spherical ellipse in the constant-sum-of-geodesic-distances sense. The correct focal angular distance for a spherical ellipse with semi-axes alpha,beta is arccos(cos alpha/cos beta), which differs from the Euclidean focal distance sqrt(alpha^2-beta^2). For alpha=pi/4, beta=pi/6 these are ~0.615 and ~0.585; at azimuth phi=pi/4 the curve defined by (7) and the true spherical ellipse are already different. So equation (7) is not an equation of a spherical ellipse, and the density contours of (2) are not shown to be spherical ellipses.\n\nThis is a load-bearing flaw: the paper's stated purpose is to provide that characterization. The reformulation alone, while correct, is a minor algebraic note. The paper needs major revision: either remove the false claim and reposition itself as a note on a coordinate-free density, or repair the geometric argument if that is possible. I would not cite the spherical-ellipse claim. For a reader working on directional statistics, the reformulation may be a convenience, but the main advertised result is wrong. A serious referee should still be assigned, because the error is subtle and the paper engages honestly with the literature; a good referee can catch it and give the authors a path to a corrected version. Recommend reject in current form.","headline":"The tangent-free rewrite of Hauberg's density is correct, but the paper's advertised spherical-ellipse characterization is false because the log map is not a global isometry.","tokens_in":7347,"tokens_out":3701,"would_cite":false,"duration_ms":37463,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H11","53C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives a tangent-space-free formula for the anisotropic geodesic normal distribution on the sphere and claims its density contours are exactly spherical ellipses.","keywords":["geodesic normal distribution","spherical normal distribution","anisotropic distribution","spherical ellipse","directional statistics","tangent space","logarithm map"],"falsifier":"Pick $\\alpha=\\pi/6$, $\\beta=\\pi/12$, set $\\gamma$ by $\\cos\\gamma=\\cos\\alpha/\\cos\\beta$, take $\\mu=(1,0,0)$ and $\\eta=(0,1,0)$, and generate points $x$ satisfying equation (7). Compare $d_G(x,f_1)+d_G(x,f_2)$ for $f_1=\\exp_\\mu(\\gamma\\eta)$ and $f_2=\\exp_\\mu(-\\gamma\\eta)$; if this sum is not constant on the curve, equation (7) does not define a spherical ellipse under definition (3).","tokens_in":6228,"feed_emoji":"🌐","tokens_out":4815,"duration_ms":43076,"temperature":0.7,"pith_summary":"This paper aims to reformulate Hauberg's anisotropic geodesic normal distribution on the sphere so that the density is defined directly on $S^{2}$, without projecting through the tangent space. The reformulation expresses the quadratic form of the log map in terms of the eigenvalues and eigenvectors of the precision matrix. The paper further claims that the density contours of this distribution are true spherical ellipses, not merely approximations, and derives a new algebraic equation for such ellipses. If these claims hold, the parameters of the distribution acquire a direct geometric meaning in terms of the major and minor axes of the contour ellipses.","feed_headline":"Geodesic normal density gets a tangent-free form","feed_subtitle":"The anisotropic spherical normal is written directly on S², with contours said to be exact spherical ellipses.","key_machinery":"The key object is the logarithm map $\\operatorname{Log}_\\mu(x)=\\frac{x-(x^\\top\\mu)\\mu}{\\sin d_G(x,\\mu)}\\arccos(x^\\top\\mu)$, which sends the sphere minus the antipodal point into the tangent space at $\\mu$. The paper uses the identity $\\operatorname{Log}_\\mu(x)^\\top\\Lambda\\operatorname{Log}_\\mu(x)=\\arccos^2(x^\\top\\mu)\\frac{\\lambda_1(x^\\top\\eta)^2+\\lambda_2(x^\\top\\xi)^2}{(x^\\top\\eta)^2+(x^\\top\\xi)^2}$ to eliminate the tangent space from the density. For the contours, the machinery is the equivalence between the level set of this quadratic form and the preimage under $\\operatorname{Log}_\\mu$ of an ellipse in the tangent space; the paper relies on $\\operatorname{Log}_\\mu$ being an isometry to transfer Euclidean focal-distance ellipses back to geodesic focal-distance ellipses on the sphere.","core_discovery":"The central claim is that Hauberg's anisotropic geodesic normal density, originally written as $f_H(x;\\mu,\\Lambda)=C_H^{-1}\\exp(-\\tfrac12 \\operatorname{Log}_\\mu(x)^\\top \\Lambda \\operatorname{Log}_\\mu(x))$, can be rewritten exactly as equation (2), a function of the three coordinates $x^\\top\\mu$, $x^\\top\\eta$, $x^\\top\\xi$ with two concentration parameters $\\lambda_1,\\lambda_2$ and no tangent-space coordinates. This identity follows from expanding $\\operatorname{Log}_\\mu(x)$ in the orthonormal basis $\\{\\mu,\\eta,\\xi\\}$. The paper then asserts that the level sets of this density, described by equation (7), are exactly ellipses on the sphere, where $\\eta$ and $\\xi$ give the axes and $\\alpha,\\beta$ are inversely related to $\\lambda_1,\\lambda_2$. It presents equation (7) as a new characterization of a spherical ellipse.","pith_inferences":["The same tangent-free substitution should extend the density formula to $S^p$ for $p>2$ by writing $\\Lambda$ with $p$ positive eigenvalues and choosing an orthonormal basis of the tangent space.","A direct numerical check of equation (7) against the focal-distance definition (3) would test whether the spherical-ellipse characterization holds for the full parameter range, especially near the antipode.","If the contour-ellipse claim holds, the geodesic normal distribution could serve as a statistical model attached to spherical ellipses, with potential uses in shape analysis and directional clustering."],"forward_implications":["The density in equation (2) allows practitioners to evaluate the geodesic normal distribution on $S^2$ without computing the log map or its inverse for each point.","The parameters gain a geometric interpretation: $\\eta$ and $\\xi$ fix the axes of the contour ellipse, and $\\lambda_1,\\lambda_2$ are inversely related to the semi-axis lengths.","Setting $\\lambda_1=\\lambda_2$ reduces the anisotropic density to the isotropic geodesic normal, giving a smooth family connecting isotropic and anisotropic models.","The claimed equivalence in equation (7) would provide a closed-form algebraic equation for spherical ellipses that is an alternative to the focal-distance definition in equation (3)."],"supporting_citations":[{"why":"Introduces the anisotropic spherical normal distribution and the log-map construction this paper reformulates.","marker":"Hauberg (2018)"},{"why":"Supplies the standard equation for spherical ellipses and the spherical Pythagorean relation used in Section 4.","marker":"Glaeser et al. (2016)"},{"why":"Defines the isotropic geodesic normal distribution on the circle and gives the normalizing constant in that case.","marker":"Coeurjolly and Bihan (2012)"},{"why":"Provides the general Riemannian normal distribution framework that motivates the tangent-space construction.","marker":"Pennec (2006)"}],"fun_headline_variants":["Spherical normal distribution freed from tangent space","Anisotropic geodesic normal gets direct sphere form","Geodesic normal on sphere with exact elliptical contours","New tangent-free form of the spherical normal","Spherical normal: density contours become ellipses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The contour-ellipse proof assumes the logarithm map is an isometry of the whole sphere, so that a Euclidean ellipse in the tangent space maps to a curve whose points have constant sum of geodesic distances to the two foci; in fact the log map preserves distances only from the pole, not between arbitrary points.","fun_headline_variants_meta":{"raw":{"variants":["Spherical normal distribution freed from tangent space","Anisotropic geodesic normal gets direct sphere form","Geodesic normal on sphere with exact elliptical contours","New tangent-free form of the spherical normal","Spherical normal: density contours become ellipses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000177,"raw_usage":{"total_tokens":1234,"prompt_tokens":828,"completion_tokens":406,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":335}},"tokens_in":444,"tokens_out":406,"duration_ms":4374,"temperature":1.0,"reasoning_tokens":335,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:47:08.715782+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick $\\alpha=\\pi/6$, $\\beta=\\pi/12$, set $\\gamma$ by $\\cos\\gamma=\\cos\\alpha/\\cos\\beta$, take $\\mu=(1,0,0)$ and $\\eta=(0,1,0)$, and generate points $x$ satisfying equation (7). Compare $d_G(x,f_1)+d_G(x,f_2)$ for $f_1=\\exp_\\mu(\\gamma\\eta)$ and $f_2=\\exp_\\mu(-\\gamma\\eta)$; if this sum is not constant on the curve, equation (7) does not define a spherical ellipse under definition (3).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the anisotropic spherical normal distribution and the log-map construction this paper reformulates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard equation for spherical ellipses and the spherical Pythagorean relation used in Section 4."},{"cited_title":"and Bihan, N","cited_arxiv_id":null,"evidence_quote":"Defines the isotropic geodesic normal distribution on the circle and gives the normalizing constant in that case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the general Riemannian normal distribution framework that motivates the tangent-space construction."}],"review_version":1}