Pith. sign in

REVIEW 3 minor 21 references

Group invariance of $f$-divergences and the Fisher--Rao distance

T0 review · 0 major / 3 minor · reviewed 2026-06-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read In transformation models, every f-divergence is invariant under the group action.

desk verdict The paper shows f-divergences are invariant under group actions in transformation models via a direct change-of-variables argument, with a clean reduction to maximal invariants or double cosets. read the letter →

arxiv 2606.25790 v1 pith:ZL3IBQWM submitted 2026-06-24 math.ST cs.ITmath.ITstat.TH

Frank Nielsen , Kazuki Okamura This is my paper · ORCID
classification math.STcs.ITmath.ITstat.TH
keywords f-divergencesFisher-Raodistancegroupinvariancetransformationmodelsmaximalinvariantsdoublecosetslocation-scalefamilies
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

The paper shows that when a statistical model admits a group action on both sample space and parameter space with densities transforming by a multiplier, every f-divergence between two distributions stays the same after the group is applied to the parameters. This invariance implies that the divergence value depends only on a maximal invariant of the parameter pair. When the group action on the parameter space is transitive, that maximal invariant reduces to a double coset. The identical reduction holds for the Fisher-Rao distance, and the authors illustrate the result on multidimensional location-scale families. A reader would care because the result lets one replace full parameter pairs with lower-dimensional invariants when comparing distributions inside symmetric families.

What carries the argument

The invariance of f-divergences under the group action in a transformation model, which reduces the divergence to a function of a maximal invariant (or double coset) of the parameter pair.

What would settle it

Compute an f-divergence for two parameter values, apply a group element to both, recompute the divergence, and check whether the two values differ inside a valid transformation model.

Watch

Extended reading notes

Core claim

We work with a transformation model in which a group acts on both the sample space and the parameter space, and the densities transform with a multiplier. Under this assumption, we show that every f-divergence is invariant under the group action. As a consequence, an invariant divergence depends only on a maximal invariant of the pair of parameters. When the action on the parameter space is transitive, this maximal invariant is given by a double coset. We apply this result to multidimensional location-scale families, and we show that the same reduction applies to the Fisher-Rao distance.

Load-bearing premise

The model must be a transformation model in which a group acts on both sample and parameter spaces and densities transform by a multiplier.

Editorial extensions

If this is right

  • Any f-divergence between two distributions equals the divergence between their group-transformed versions.
  • When the group action is transitive the divergence depends only on the double coset of the parameter pair.
  • The Fisher-Rao distance admits the same invariance and therefore the same reduction to maximal invariants.
  • In multidimensional location-scale families the divergences and the Fisher-Rao distance can be expressed using only the invariants of the location and scale parameters.

Reading between the lines

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

  • Numerical evaluation of divergences inside symmetric families can be performed on a reduced parameter space of maximal invariants.
  • Any statistical procedure that relies on an f-divergence or the Fisher-Rao distance can be made group-equivariant by working directly with the maximal invariant.
  • The same invariance argument may extend to other information-geometric objects that are defined via integrals against the densities.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The manuscript shows that, in a transformation model where a group G acts on both the sample space and the parameter space such that densities transform by the same multiplier, every f-divergence between two distributions is invariant under the group action. Consequently any such invariant divergence depends only on a maximal invariant of the pair of parameters; when the action on the parameter space is transitive this maximal invariant is realized by a double coset. The reduction is applied to multidimensional location-scale families and is shown to hold for the Fisher–Rao distance as well.

Significance. If the central invariance argument holds, the paper supplies a clean, assumption-light reduction that lets invariant divergences and the Fisher–Rao metric be expressed solely in terms of maximal invariants (or double cosets). This is a useful structural result for any symmetric model and is particularly concrete for location-scale families, where it can simplify both theoretical comparisons and numerical work.

minor comments (3)
  1. [§3] §3, after the statement of the main invariance theorem: the change-of-variables argument for the integral is only sketched; writing the explicit substitution x ↦ g·x and the cancellation of the common multiplier would make the step fully self-contained.
  2. [§5] §5 (location-scale application): the double-coset description is stated but no concrete coordinate chart or reduced expression for a standard f-divergence (e.g., KL) is supplied; a short worked example would clarify the practical gain.
  3. Notation: the symbol for the dominating measure is introduced only in the transformation-model definition and then used without re-statement in later sections; a single sentence recalling that the measure is G-quasi-invariant would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The summary provided accurately captures the main contributions.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation self-contained from transformation model definition

full rationale

The paper establishes invariance of every f-divergence under the group action directly from the transformation model assumption (densities acquire identical Jacobian multipliers). This follows by change-of-variables in the integral definition of the f-divergence; the ratio of densities is preserved and the integral is invariant. The further reduction to a maximal invariant (or double coset) is immediate from the definition of group invariance and does not rely on fitted quantities, self-citations, or smuggled ansatzes. The application to location-scale families and the Fisher-Rao distance is likewise a direct specialization. No load-bearing step reduces to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the domain assumption of a transformation model with group action and density multiplier; no free parameters or invented entities are indicated in the abstract.

assumptions (1)
  • domain assumption The statistical model is a transformation model in which a group acts on both the sample space and the parameter space, and the densities transform with a multiplier.
    This is the explicit setup stated in the abstract under which the invariance of every f-divergence is shown.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Group invariance of $f$-divergences and the Fisher--Rao distance." pith.science (2026). https://pith.science/paper/ZL3IBQWM

@misc{pith2026260625790,
  author       = {Pith},
  title        = {Pith review of: Group invariance of $f$-divergences and the Fisher--Rao distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZL3IBQWM}},
  note         = {Machine review of arXiv:2606.25790}
}
abstract

Many statistical models have natural symmetries described by a group action. We study how such symmetries affect the comparison of two distributions. We work with a transformation model in which a group acts on both the sample space and the parameter space, and the densities transform with a multiplier. Under this assumption, we show that every $f$-divergence is invariant under the group action. As a consequence, an invariant divergence depends only on a maximal invariant of the pair of parameters. When the action on the parameter space is transitive, this maximal invariant is given by a double coset. We apply this result to multidimensional location-scale families, and we show that the same reduction applies to the Fisher--Rao distance.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references

  1. [1]

    PhD thesis, The University of Texas at Austin, 2013

    Sreangsu Acharyya.Learning to Rank in Supervised and Unsupervised Settings using Con- vexity and Monotonicity. PhD thesis, The University of Texas at Austin, 2013

    2013

  2. [2]

    A general class of coefficients of divergence of one distribution from another.Journal of the Royal Statistical Society: Series B (Methodologi- cal), 28(1):131–142, 1966

    Syed Mumtaz Ali and Samuel D Silvey. A general class of coefficients of divergence of one distribution from another.Journal of the Royal Statistical Society: Series B (Methodologi- cal), 28(1):131–142, 1966

    1966

  3. [3]

    Applied Mathematical Sci- ences

    Shun-ichi Amari.Information Geometry and Its Applications. Applied Mathematical Sci- ences. Springer Japan, 2016

    2016

  4. [4]

    Springer-Verlag, New York,

    Christian Berg, Jens Peter Reus Christensen, and Paul Ressel.Harmonic analysis on semigroups, volume 100 ofGraduate Texts in Mathematics. Springer-Verlag, New York,

  5. [5]

    Theory of positive definite and related functions

  6. [6]

    An explicit solution of information geodesic equations for the multivariate normal model.Statistics & Risk Modeling, 9(1-2):119–138, 1991

    Miquel Calvo and Josep Maria Oller. An explicit solution of information geodesic equations for the multivariate normal model.Statistics & Risk Modeling, 9(1-2):119–138, 1991

    1991

  7. [7]

    Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten.Magyar Tud

    Imre Csiszár. Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten.Magyar Tud. Akad. Mat. Kutató Int. Közl., 8:85–108, 1963

    1963

  8. [8]

    Eaton.Group invariance applications in statistics, volume 1 ofNSF-CBMS Re- gional Conference Series in Probability and Statistics

    Morris L. Eaton.Group invariance applications in statistics, volume 1 ofNSF-CBMS Re- gional Conference Series in Probability and Statistics. Institute of Mathematical Statistics, Hayward, CA; American Statistical Association, Alexandria, VA, 1989

    1989

Show all 21 references
  1. [9]

    Geometry of minimum contrast.Hiroshima Mathematical Journal, 22(3):631–647, 1992

    Shinto Eguchi. Geometry of minimum contrast.Hiroshima Mathematical Journal, 22(3):631–647, 1992

    1992

  2. [10]

    Springer Science & Business Media, 2010

    Robert W Keener.Theoretical statistics: Topics for a core course. Springer Science & Business Media, 2010

    2010

  3. [11]

    Geodesics of multivariate normal distributions and a Toda lattice type Lax pair.Physica Scripta, 98(11):115241, 2023

    Shimpei Kobayashi. Geodesics of multivariate normal distributions and a Toda lattice type Lax pair.Physica Scripta, 98(11):115241, 2023. 14

    2023

  4. [12]

    Multivariate normal distributions parametrized as a Riemannian symmetric space.Journal of Multivariate Analysis, 74(1):36– 48, 2000

    Miroslav Lovrić, Maung Min-Oo, and Ernst A Ruh. Multivariate normal distributions parametrized as a Riemannian symmetric space.Journal of Multivariate Analysis, 74(1):36– 48, 2000

    2000

  5. [13]

    Markov processes and the H-theorem.J

    Tetsuzo Morimoto. Markov processes and the H-theorem.J. Phys. Soc. Japan, 18:328–331, 1963

    1963

  6. [14]

    Approximation and bounding techniques for the Fisher-Rao distances be- tween parametric statistical models

    Frank Nielsen. Approximation and bounding techniques for the Fisher-Rao distances be- tween parametric statistical models. InHandbook of Statistics, volume 51, pages 67–116. Elsevier, 2024

    2024

  7. [15]

    On thef-divergences between densities of a multi- variate location or scale family.Stat

    Frank Nielsen and Kazuki Okamura. On thef-divergences between densities of a multi- variate location or scale family.Stat. Comput., 34(1):Paper No. 60, 11, 2024

    2024

  8. [16]

    Geometry on positive definite matrices induced from V-potential function

    Atsumi Ohara and Shinto Eguchi. Geometry on positive definite matrices induced from V-potential function. InGeometric science of information. First international conference, GSI 2013, Paris, France, August 28–30, 2013. Proceedings, pages 621–629. Berlin: Springer, 2013

    2013

  9. [17]

    Geometry on positive definite matrices deformed by V-potentials and its submanifold structure

    Atsumi Ohara and Shinto Eguchi. Geometry on positive definite matrices deformed by V-potentials and its submanifold structure. InGeometric theory of information. Selected and revised contributions of the first conference on the geometric sciences of information, GSI, Paris, Fr...

    2013

  10. [18]

    Dualistic differential geometry of positive definite matrices and its applications to related problems.Linear Algebra and its Applications, 247:31–53, 1996

    Atsumi Ohara, Nobuhide Suda, and Shun-ichi Amari. Dualistic differential geometry of positive definite matrices and its applications to related problems.Linear Algebra and its Applications, 247:31–53, 1996

    1996

  11. [19]

    The Fisher–Rao distance be- tween multivariate normal distributions: Special cases, bounds and applications.Entropy, 22(4):404, 2020

    Julianna Pinele, João E Strapasson, and Sueli IR Costa. The Fisher–Rao distance be- tween multivariate normal distributions: Special cases, bounds and applications.Entropy, 22(4):404, 2020

    2020

  12. [20]

    Information and the accuracy attainable in the estimation of statistical parameters.Bulletin of the Calcutta Mathematical Society, 37(3):81–91, 1945

    Calyampudi Radhakrishna Rao. Information and the accuracy attainable in the estimation of statistical parameters.Bulletin of the Calcutta Mathematical Society, 37(3):81–91, 1945

    1945

  13. [21]

    Wijsman.Invariant Measures on Groups and Their Use in Statistics, volume 14 ofInstitute of Mathematical Statistics Lecture Notes–Monograph Series

    Robert A. Wijsman.Invariant Measures on Groups and Their Use in Statistics, volume 14 ofInstitute of Mathematical Statistics Lecture Notes–Monograph Series. Institute of Math- ematical Statistics, Hayward, CA, 1990. 15

    1990

Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.