Geometric Information Decomposition for Weighted Empirical Measures on the Sphere

Boram Cho; Kisung You

Nested maximum-entropy gaps on the sphere turn a single concentration number into a geometric uncertainty profile that separates mean-direction, axial, and multimodal structure.

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2026-07-12 05:55 UTC pith:ARA7QIE6

load-bearing objection Solid operational packaging of classical nested max-ent on the sphere, with usable null calibration and experiments that match the theory.

arxiv 2607.02943 v3 pith:ARA7QIE6 submitted 2026-07-03 stat.ME

Geometric Information Decomposition for Weighted Empirical Measures on the Sphere

Kisung You , Boram Cho This is my paper

classification stat.ME MSC 62H1162B1094A17

keywords geometric information decompositiondirectional statisticsvon Mises–FisherFisher–Binghamspherical harmonicsmaximum entropyentropy gapsweighted empirical measures

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

When data are already a weighted cloud of directions on the unit sphere, the usual von Mises–Fisher fit only sees mean-direction strength and can report near-uniform uncertainty even when the mass is tightly concentrated on an axis, a girdle, or several symmetric modes. This paper defines the geometric information decomposition: a nested sequence of maximum-entropy densities that match successive spherical moments, then reports the entropy gap each new level adds. Level 1 recovers ordinary vMF information; level 2 captures Fisher–Bingham-type anisotropy; higher harmonic levels pick up residual multimodality. The gaps are rotation- and basis-invariant, consistent, and equipped with a second-order quadratic-form test for whether a new level truly carries information. Controlled circle and sphere examples, null-calibration checks, and a query-weighted digit projection show when the first gap is enough and when later gaps expose structure that a single concentration number would miss.

Core claim

For a weighted empirical measure on the sphere, the entropy increments I_L = D_L − D_{L−1} of nested maximum-entropy projections onto successive spherical feature spaces equal the KL distance between consecutive projections; they recover vMF information at level 1, Fisher–Bingham-type axial and girdle structure at level 2, and finer angular multimodality at higher levels, with invariance, consistency, normal limits away from zero, and a quadratic-form null calibration for testing new levels.

What carries the argument

Geometric information decomposition (GID): the profile of entropy gaps I_L(P) = KL(p_L^P ν ∥ p_{L−1}^P ν) arising from nested maximum-entropy projections that match successive spherical (or harmonic) moments; the first gap is vMF information and the second is Fisher–Bingham-type anisotropy.

Load-bearing premise

The observed moments must sit strictly inside the feasible moment body so that finite natural parameters exist; if they hit the boundary the unregularized gaps are no longer well-defined in the usual way.

What would settle it

On a controlled antipodal or girdle sample whose mean resultant length is near zero, the second information gap should be large and the first near zero; if both gaps remain near zero (or if the quadratic-form null calibration rejects uniformity when the sample is truly uniform), the claimed geometric separation fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Solid operational packaging of classical nested max-ent on the sphere, with usable null calibration and experiments that match the theory.

read the letter

This is a clean, useful diagnostic paper, not a foundational rewrite. The core identity is classical hierarchical KL (Csiszar/Amari), and the author says so. What is new is the packaging: treat weighted empirical measures on the sphere (importance samples, attention weights, quadrature) as the object, report the successive entropy gaps as a geometric fingerprint, give sphere-specific readings of I1 (vMF) and I2 (Fisher–Bingham/axial/girdle), and supply the second-order quadratic-form null calibration (Theorem 8) with a sandwich recipe that actually works under nonuniform weights.

The math is standard exponential-family and delta-method work, written carefully. Invariance, consistency, alternative-regime normality, and the null chi-bar / quadratic-form limit are all there; the local expansions for I1 and I2 match the usual moment calculations. Experiments are modest but on-target: Fourier hierarchy on S1 recovers the intended dominant gaps for vMF / antipodal / trimodal; low-order S2 examples separate mean-direction from axial/girdle structure; the null-calibration table and ECDFs show that naive chi-squared fails under importance weights while the plug-in quadratic form does not. Code is linked. Residual diagnostics and ridge regularization for boundary moments are discussed honestly.

Soft spots are real but proportional. The relative-interior moment assumption is load-bearing for the main theorems; boundary cases need regularization whose effect on the reported gaps is only sketched. Full harmonic fitting does not scale to high ambient dimension, which the paper acknowledges and mitigates with structured/projected features. Order selection is left open. None of that breaks the claims inside the moderate-dimensional or structured-feature regime the author targets.

This is for people who already work with directional data, spherical embeddings, or weighted particles and who currently report only kappa or mean resultant length. It deserves a serious referee. I would engage with it and expect to cite the gap profile and the null recipe when I next need a directional uncertainty diagnostic beyond vMF.

Referee Report

0 major / 5 minor

Summary. The paper proposes geometric information decomposition (GID) for weighted empirical measures on the sphere (or compact manifolds). Nested maximum-entropy projections onto successive spherical feature spaces yield entropy deficits D_L and gaps I_L = D_L - D_{L-1} = KL(p_L || p_{L-1}). Level 1 recovers vMF mean-direction information, level 2 Fisher–Bingham/Bingham-type anisotropy, and higher levels finer harmonic structure. The authors prove basis and isometry invariance, consistency, delta-method normality away from zero gaps, and a second-order quadratic-form null calibration (Theorems 1–8), with local expansions of I_1 and I_2. Experiments on S^1 and S^2 recover the intended dominant gaps, validate null calibration under equal and importance weights, and illustrate a query-weighted digit projection.

Significance. If the results hold, GID supplies a principled, interpretable alternative to reporting only vMF concentration for weighted directional data (importance samples, quadrature, attention-weighted embeddings). The hierarchy cleanly separates mean-direction, axial/girdle, and multimodal structure that a single first-moment fit misses. Strengths include explicit specialization of classical hierarchical KL identities rather than a claimed new identity, complete asymptotic theory with an operational null recipe (Theorem 8 and plug-in covariance), local moment expansions matching known calculations, and reproducible experiments with code. The relative-interior moment assumption is standard exponential-family practice and is acknowledged with regularization and residual diagnostics. The contribution is a useful, well-calibrated diagnostic profile for moderate-dimensional directional settings and structured high-d variants.

minor comments (5)

Section 7 and Appendix B.3 correctly flag boundary moments and ridge regularization, but a short numerical illustration of how small alpha affects reported I_L near the boundary would help practitioners decide when regularization is safe for the descriptive profile.
Table 1 and Table 2 report Monte Carlo SEs over replicates; adding the corresponding effective sample sizes more prominently in the table captions (already in the text) would make weight effects easier to read at a glance.
Figure 1 caption and the tetrahedral panel are clear; a brief note that the orthographic projection can visually compress antipodal pairs would avoid any misreading of the point clouds.
Appendix B.1 residual (9): stating the default a_ell and K_max used in the main S^1 residual numbers (already given in the text) in the equation caption would improve self-containment.
A few minor typos appear (e.g., spacing around bR, bkappa in early sections; 'Garc´ıa-Portugu´es' accent rendering). These are purely presentational.

Circularity Check

0 steps flagged

No significant circularity: classical hierarchical KL specialized to spherical features with independent estimation theory and experiments.

full rationale

The paper's central structural identity (Theorem 1) is explicitly presented as a specialization of classical nested maximum-entropy / Pythagorean KL decompositions (Csiszar, Amari et al.), not as a novel derivation from first principles. The gaps I_L are defined as entropy differences of successive max-ent projections that match empirical moments of chosen feature spaces; they are estimated by dual likelihood on those moments and are not re-labeled fitted constants. Invariance, consistency, delta-method normality away from zero, and the second-order quadratic-form null calibration (Theorems 2–8) follow standard exponential-family arguments under the relative-interior moment assumption, which the paper states and handles with regularization diagnostics. Experiments compare the gaps to classical summaries (R-bar, kappa, Frobenius anisotropy) on controlled mixtures and a projected digit example; no quantity is fitted to data and then presented as an independent prediction of a closely related quantity. Self-citations are absent; related-work citations are external. The only minor definitional element is that level-1 recovers vMF by construction of the linear feature space, which the paper states openly rather than claiming as an empirical discovery. Overall circularity burden is negligible.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 1 invented entities

The central claims rest on classical maximum-entropy / exponential-family geometry specialized to nested spherical features and on standard moment LLN/CLT under weights. No new physical entities; free choices are feature spaces, optional regularization, and experimental hyperparameters. The load-bearing modeling assumptions are interior moments and nested continuous mean-zero features.

free parameters (4)

feature spaces V_L (harmonic degree or structured subspace)
User-chosen nested spaces determine which structure is attributed to each gap; full harmonics vs low-rank/sketch features change interpretation while preserving formal KL-gap theory.
ridge regularization strength alpha (when used)
Appears in dual when moments approach the boundary; shifts fitted moments and thus reported gaps; not fixed by theory.
Sobolev residual weights a_ell and truncation K_max
Diagnostic hyperparameters that affect residual size R_L used to judge adequacy of a level.
softmax temperature tau=6 (digit example)
Controls query weights in the real-data illustration; changes effective sample size and gap magnitudes.

axioms (6)

domain assumption M is a connected compact Riemannian manifold without boundary; nu is normalized Riemannian volume; densities w.r.t. nu.
Setup section; supplies uniform reference and avoids integrability issues for the entropy deficits.
domain assumption Nested finite-dimensional continuous mean-zero feature spaces V_0 subset ... subset V_L in L2_0(M,nu).
Defines the hierarchy; nesting is required for monotonicity and the KL-gap identity (Theorem 1).
domain assumption Relevant moment vectors m_L(P) lie in the relative interior of the moment body M_L.
Assumption 4 and main statements; guarantees existence of finite natural parameters and differentiability of D_L.
standard math Standard finite-dimensional exponential-family theory: regular/steep families, mean map diffeomorphism onto ri(M), uniqueness of max-ent projection.
Cited via Brown (1986), Csiszar–Matus; used for existence (Appendix C.1) and dual representation.
domain assumption Moment convergence (LLN) and, for asymptotics, a CLT a_n (bm_L - m_L) => N(0, Sigma_L) under the sampling/weighting mechanism.
Assumptions 5 and display (5); covers empirical, deterministic weights, importance samples when the corresponding limit holds.
standard math Classical hierarchical KL / Pythagorean identity for nested exponential families.
Theorem 1 is explicitly a specialization of Csiszar/Amari; paper does not claim a new abstract identity.

invented entities (1)

Geometric information decomposition (GID) profile (I_1,...,I_L) no independent evidence
purpose: Operational uncertainty fingerprint that attributes non-uniform information to successive spherical feature levels for weighted empirical measures.
The named diagnostic and its estimation/calibration package; independent evidence is experimental recovery of known geometries and null calibration, not an external physical prediction.

pith-pipeline@v1.1.0-grok45 · 20389 in / 3171 out tokens · 32542 ms · 2026-07-12T05:55:03.883477+00:00 · methodology

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read the original abstract

Weighted observations on the unit sphere arise in importance sampling, quadrature, and attention-weighted embeddings. Directional uncertainty is often summarized through a von Mises-Fisher (vMF) fit and its concentration or entropy. This summary uses only mean-direction information. It can miss antipodal, axial, girdle-like, or multimodal structure. We introduce geometric information decomposition (GID), which fits nested maximum-entropy projections to spherical features. Each gap measures the entropy reduction contributed by one feature level. The first gap is the fitted vMF distribution's KL divergence from uniformity. The second measures residual quadratic information, including Fisher-Bingham anisotropy. Later gaps describe finer angular structure. We establish invariance, consistency, alternative-regime asymptotic normality, and quadratic-form null calibration. Circular and spherical experiments include importance-weight calibration and a query-weighted digit projection. The results separate settings where vMF uncertainty is adequate from settings with higher-order structure.