M\"obius transport on spheres

Eduargo Garc\'ia-Portugu\'es; Shogo Kato

REVIEW 1 major objections 1 minor 22 references

M\"obius transport on spheres

T0 review · 1 major / 1 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The Möbius map that turns the uniform law on a sphere into the spherical Cauchy law is an optimal transport of the cosine coordinate.

desk verdict Möbius transport as optimal transport on spheres is a clean, real result; the paleomagnetic application has a genuine preprocessing soft spot, but the core theory holds and deserves peer review. read the letter →

arxiv 2607.29280 v1 pith:QT5YKJMW submitted 2026-07-31 math.ST stat.MEstat.TH

Eduargo García-Portugués , Shogo Kato This is my paper · ORCID

classification math.STstat.MEstat.TH MSC 62H1149Q22

keywords MöbiustransformationoptimaltransportsphericalCauchydistributionvonMises–Fisherrotationalsymmetrymonotonerearrangementdirectionalstatisticsdistributions

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

This paper shows that a classical device in directional statistics—the Möbius transformation that turns the uniform law on the sphere into the spherical Cauchy law—is an optimal transport in disguise: it moves probability mass by monotonically rearranging the cosine of each point about the location axis and leaving the tangent direction alone. Because this rearrangement is determined entirely by the target's projected cumulative distribution function, any rotationally symmetric law can replace the Cauchy target, producing a generalized Möbius transport. Transporting a von Mises–Fisher base by such a map gives anisotropic spherical distributions with explicit densities, inherited normalizing constants, and exact simulation. The resulting models fit paleomagnetic directions and short-period comet orbit normals better than classical alternatives, capturing skewness and heavy tails that standard models miss.

What carries the argument

The central object is the tangent–normal lift of the monotone rearrangement of the cosine. For a rotationally symmetric target G about a location axis μ, its projected cumulative distribution function F_G defines a map t ↦ F_G^{-1}(F_0(t)) on the cosine coordinate t = x^T μ; applying this map while preserving the tangent vector gives M_G. This single construction is the optimal transport from the uniform to G, specializes to the classical Möbius transformation when G is the spherical Cauchy law, and converts a von Mises–Fisher base into an anisotropic transported distribution with closed-form density.

What would settle it

Re-fit the paleomagnetic data on the normal and reversed polarities separately, or with an alternative polarity-reversal criterion; if the Möbius–von Mises–Fisher model no longer achieves the lowest AIC/BIC, the empirical conclusion is an artifact of the merging step. A second check: compute the Jacobian of the generalized map for a Poisson-kernel target on S^2 at a point with cosine not ±1; if it is conformal, the uniqueness claim for the spherical Cauchy target is false.

Watch

Extended reading notes

Core claim

The paper establishes that the Möbius transformation generating the spherical Cauchy distribution from the uniform on the sphere is exactly the tangent–normal lift of a one-dimensional optimal transport: the monotone rearrangement of the cosine about the location axis, expressed as F_ρ = F_0 ∘ M_ρ. Since this lift depends on the target only through its projected cumulative distribution function, replacing the Cauchy target by any rotationally symmetric law with a continuous, strictly increasing projected cdf produces a generalized Möbius transformation M_G that carries the uniform to G. Transporting a von Mises–Fisher base by such a map yields anisotropic spherical distributions with closed-

Load-bearing premise

The empirical outperformance claim rests on the preprocessing step that reflects reversed-polarity paleomagnetic directions to a common polarity; if that merge is not a faithful representation of a single underlying population, the reported skewness and the AIC/BIC ranking of Möbius–von Mises–Fisher over competitors could be artifacts.

Editorial extensions

If this is right

If the paper is right, the spherical Cauchy map is an explicit optimal transport on the sphere, and the spherical Cauchy law is the unique conformal member of a broader family of cosine-rearranging maps.
Any rotationally symmetric target with a continuous, strictly increasing projected cdf—such as the Poisson kernel or spherical cardioid—yields a tractable generalized Möbius transformation with closed-form map and exact simulation.
Möbius-transported von Mises–Fisher distributions give anisotropic spherical models whose normalizing constants are fully explicit, avoiding the infinite-series constants of the classical Kent distribution.
The isotropic scaled von Mises–Fisher distribution is identified as a Möbius transport, explaining its good fits to paleomagnetic and comet data within the same framework.
Composing generalized Möbius transformations with different axes produces skewed or bimodal spherical densities in closed form, opening a route to multi-axis models.

Reading between the lines

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

Because the transport depends only on the target's projected cdf, one could plug in an empirical cdf estimate to define nonparametric spherical transport maps, potentially yielding distribution-free ranks or tests for directional data—an extension the paper does not explore.
The tangent–normal split suggests a general recipe: any monotone transform of the cosine coordinate lifts to a spherical map, so the construction might serve as a copula-style device that separates the cosine marginal from the uniform tangent structure.
The paper notes but does not pursue compositions M_{G_p} ∘ ... ∘ M_{G_1} with distinct axes; fitting such compositions to bimodal directional data is a direct next step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core idea is genuinely new and clean: writing the spherical Cauchy Möbius map as the lift of a one-dimensional optimal transport (Prop 2.2, Eq. (2)) is a real reframing, not a cosmetic one. It turns an analytic transformation into a measure-transport statement, and the generalization to any rotationally symmetric target (Section 3) follows naturally. The density formulas (7)-(10) are explicit, closed-form, and the simulation scheme is immediate. This is a solid contribution to directional statistics.

What the paper does well: the proofs are careful and readable. The tangent–normal decomposition is used honestly, the density transform in Theorem 3.1 follows from a plain pushforward argument, and the closed-form quantiles check out. The identification of the isotropic scaled von Mises–Fisher as a Möbius transport (Prop 4.1) is a nice unification. I believe the math. The paper is self-contained and does not rely on circular reasoning.

The soft spots are mostly at the application level. The paleomagnetic comparison depends on reflecting reversed-polarity directions into a common polarity. The stress-test note worries that two nearly antipodal modes 3.2° apart could be two populations, and the MvMF's skewness axis could absorb the between-mode offset. That concern is legitimate and not manufactured. The paper does not check the two polarity groups separately, nor does it compare against a two-component mixture. So the 'outperform' claim on paleomagnetic data rests on the merging assumption. But that is an application-level assumption, not a flaw in the theory. The comet application is cleaner and the scaled vMF fit there is honestly evaluated with confidence bands.

The only theoretical gap I notice is the asserted conformal-uniqueness ODE solution in Section 3 (the claim that only Möbius maps solve ψ'(t) = (1−ψ(t)^2)/(1−t^2)). They state it without proof; it looks like a standard Riccati argument, and it is minor, but worth a line in the paper.

Who the paper is for: anyone working on directional distributions, particularly anisotropic models with explicit normalizing constants. The Möbius–von Mises–Fisher is a practical alternative to Kent with a closed-form normalizer, and the OT framing gives it conceptual depth. I would bring it to a reading group and I would cite it. It deserves peer review. The paleomagnetic section needs a robustness check or a softened claim, but the main result is solid.

Recommendation: send it to a serious referee. Expect minor revision on the application section and the uniqueness proof; the theory stands.

Referee Report

1 major / 1 minor

Summary. The paper shows that the Möbius map generating the spherical Cauchy distribution from the uniform law is the tangent–normal lift of a one-dimensional optimal transport: it acts on the cosine coordinate by the monotone rearrangement F_0^{-1}∘F_ρ and fixes the tangent direction. Proposition 2.2 and Corollary 2.1 formalize this. The authors then generalize the construction by replacing the Cauchy target with any rotationally symmetric law with continuous strictly increasing projected cdf, obtaining the "generalized Möbius transformation" M_G. Theorem 3.1 gives a density-transform formula, and Section 3.1 provides closed-form cosine quantile functions for the Poisson kernel and spherical cardioid targets. Section 4 constructs two tractable anisotropic families, the Möbius–von Mises–Fisher and isotropic scaled von Mises–Fisher distributions, with closed-form densities, normalizing constants inherited from vMF, and exact simulation. The empirical sections fit these models to paleomagnetic directions and short-period comet orbit normals, reporting improvements in AIC/BIC over classical and recent alternatives.

Significance. The optimal-transport interpretation of the spherical Cauchy Möbius map is conceptually valuable and, to my knowledge, new. It explains why the Möbius map moves mass along meridians and isolates the role of the projected cdf. The generalization to arbitrary rotationally symmetric targets is elegant and productive: Theorem 3.1's density formula is simple and correct, and the resulting MvMF distribution has an explicit normalizing constant, in contrast to Kent's distribution. The closed-form quantile maps in Corollary 3.1 and the exact simulation schemes are concrete strengths. If the mathematical core is accepted, the paper gives a useful toolkit for anisotropic spherical distributions. The empirical demonstrations are suggestive, but their strength is limited by the data-preprocessing issue discussed in the major comments.

major comments (1)

[References] The manuscript cites 'García-Portugués (2026)' and the sphunif package version 1.4.4 as available works. If these are not yet published, please mark them as in press/under review, since the spherical cardioid distribution is used as a building block.

minor comments (1)

[Section 1] The phrasing 'Möbius transformations are a recurring device' is a bit general; consider naming specifically the spherical Cauchy and wrapped Cauchy examples at the start to set the context.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the OT identification and transported-family construction are derived from stated definitions, and the empirical comparisons are in-sample fits rather than fitted-parameter predictions.

full rationale

The paper's central derivation is self-contained. Proposition 2.1 derives the projected Cauchy cdf as Fρ = F0 ∘ Mρ by a change of variables; Proposition 2.2 then uses this identity to write Mρ = F0^{-1} ∘ Fρ, which is exactly the monotone rearrangement, with optimality following from the cited standard theorem (Santambrogio 2015, Theorem 2.9). No parameter is fitted and then renamed as a prediction; the map is derived, not estimated. The generalized Möbius transformation MG is defined as the tangent-normal lift of F_G^{-1}∘F0, so the statement MG(U) ∼ G is true by construction—an explicit definition, not a circular prediction. Theorem 3.1's density formula follows from a direct pushforward computation and the stated base density. Proposition 4.1 identifies the isotropic scaled vMF as a member of the transported family by showing its scaling map has the same cosine-rearrangement structure; this is a mathematical identification, not a renaming of an empirical pattern. The applications fit models to data and compare AIC/BIC; these are in-sample model comparisons, not out-of-sample predictions from fitted constants, so they are not circular in the sense of this analysis. The paleomagnetic polarity-reflection preprocessing noted in Section 5.1 is a substantive domain assumption that could affect the empirical ranking, but that is a data-validity concern, not a derivation-circularity concern. Self-citations (spherical cardioid, sphunif, tangent vMF, Kato–McCullagh) supply external building blocks, datasets, or known distributions; they are not used to force the central result. No circular step was found.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The theoretical core is built on standard OT and tangent-normal factorization results plus explicit regularity assumptions on the target's projected cdf (FG continuous and strictly increasing). The empirical sections add data-preprocessing and i.i.d. assumptions. The listed 'free parameters' are ordinary ML estimates in the applications, not hidden degrees of freedom in the derivations. No new physical entities are postulated.

free parameters (3)

MvMF paleomagnetic MLEs (κ, ρ, axis separation) = κ=5.55, ρ=0.51, axis separation 72°
Section 5.1: fitted by maximum likelihood for the illustrative comparison; not part of the theoretical construction.
Isotropic scaled vMF comet MLEs = not reported (4-parameter model)
Section 5.2: fitted by maximum likelihood; the model's AIC/BIC win depends on these fitted values, which the paper does not list.
KDE bandwidth in Figure 2 = rule-of-thumb (not numeric)
Used only for the kernel density estimate in the paleomagnetic illustration; not load-bearing.

assumptions (6)

standard math Monotone rearrangement between atomless laws on R is the unique optimal transport map for any cost h(s−t) with h strictly convex (Santambrogio 2015, Thm 2.9).
Used in Proposition 2.2 to identify Mρ as the OT map; cited to Santambrogio.
standard math A rotationally symmetric law on S^d factorizes as independent cosine marginal and uniform S^{d−1} in the tangent–normal decomposition.
Used in Corollary 2.1 and all density derivations to reduce spherical transport to a one-dimensional cosine rearrangement.
domain assumption The target law G used in the generalized Möbius transformation has a continuous, strictly increasing projected cdf FG and a density h_G(y^Tµ)/ω_d.
The paper restricts to such targets (Section 3) so that M_G is a bijection and the density transform (3) is valid.
standard math The only increasing solutions of ψ'(t) = (1−ψ(t)^2)/(1−t^2) are Möbius maps M_{−ρ}.
Section 3's conformal-uniqueness claim; asserted rather than proved, but follows from standard ODE solution via atanh.
domain assumption The paleomagnetic data after PmagDiR::common_DI() reflection form a single population for likelihood comparison.
Section 5.1; if invalid, the application comparison and the 'outperform' claim for paleomagnetic data would be compromised.
domain assumption Orbit normals of non-fragment short-period comets are i.i.d. and rotationally symmetric (not rejected in prior tests).
Section 5.2; the fitted models are rotationally symmetric; violation could affect the reported ranking.

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Cite this review

Pith. "Pith review of M\"obius transport on spheres." pith.science (2026). https://pith.science/paper/QT5YKJMW

@misc{pith2026260729280,
  author       = {Pith},
  title        = {Pith review of: M\"obius transport on spheres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QT5YKJMW}},
  note         = {Machine review of arXiv:2607.29280}
}

read the original abstract

The M\"obius transformation that generates the spherical Cauchy distribution from the uniform is the tangent-normal lift of a one-dimensional optimal transport: the monotone rearrangement of the cosine about the location axis. This identifies the probabilistic nature of the M\"obius transformation and suggests a generalization: replacing the Cauchy target by any rotationally symmetric law, for instance the Poisson kernel or spherical cardioid, yields a generalized M\"obius transformation. M\"obius transport of a von Mises-Fisher base gives tractable anisotropic distributions on the sphere, with closed-form densities that inherit the base normalizing constant and allow immediate simulation. The M\"obius-von Mises-Fisher and isotropic scaled von Mises-Fisher distributions, the latter also arising from a M\"obius transport, are illustrated on paleomagnetic directions and short-period comet orbits, where they outperform classical and recently proposed alternatives.

Figures

Figures reproduced from arXiv: 2607.29280 by the authors.

**Figure 1.** Densities (7) on S 2 generated from a vMF(µ1 , 3) base (see Section 4.1) by the generalized Möbius transformation with target G = vMF(µ2 , λ). Hammer projections are centred on µ1 (+). White dashed curves are source cosine levels about µ2 (×); orange curves and arrows show their transported images. Each panel has its own square-root colour scale. A dual mechanism for spherical asymmetry leaves the distribution of th… view at source ↗

**Figure 2.** ). Only the parsimonious isotropic scaled von Mises–Fisher, itself a transport member (Proposition 4.1), is preferred by BIC. (a) Kernel density estimate. 0 1 2 3 (b) Fitted density (8) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Short-period comet orbit normals: empirical projected cdf and projected cdfs of the four fitted models, expressed in terms of the inclination i and shown on a square-root axis. Shading is a simultaneous 95% Kolmogorov–Smirnov band; dashed grey curves are pointwise 95% bands. Acknowledgments Financial support from grant PID2024-158399NB-I00, funded by MICIU/AEI/10.13039/501100011033 and ERDF/EU, is acknowledged. A Pr… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 1 linked inside Pith

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. , 44(4):375--417

1991

[2] [2]

Dallanave, E. (2024). Assessing the reliability of paleomagnetic datasets using the R package PmagDiR . Sci. Rep. , 14:1666

2024

[3] [3]

Dallanave, E., Agnini, C., Muttoni, G., and Rio, D. (2012). Paleocene magneto-biostratigraphy and climate-controlled rock magnetism from the B elluno B asin, T ethys O cean, I taly. Palaeogeogr. Palaeoclimatol. Palaeoecol. , 337--338:130--142

2012

[4] [4]

Downs, T. D. and Mardia, K. V. (2002). Circular regression. Biometrika , 89(3):683--698

2002

[5] [5]

Garc\'ia-Portugu\'es, E. (2013). Exact risk improvement of bandwidth selectors for kernel density estimation with directional data. Electron. J. Stat. , 7:1655--1685

2013

[6] [6]

Garc\'ia-Portugu\'es, E. (2026). On the spherical cardioid distribution and its goodness-of-fit. arXiv:2601.16095

arXiv 2026

[7] [7]

Garc\'ia-Portugu\'es, E., Paindaveine, D., and Verdebout, T. (2020). On optimal tests for rotational symmetry against new classes of hyperspherical distributions. J. Am. Stat. Assoc. , 115(532):1873--1887

2020

[8] [8]

Garc\'ia-Portugu\'es, E., Paindaveine, D., and Verdebout, T. (2025). On a class of S obolev tests for symmetry, their detection thresholds, and asymptotic powers. J. Am. Stat. Assoc

2025