Weyl symmetry of the gradient-flow in information geometry

Sousuke Noda; Tatsuaki Wada

REVIEW 3 major objections 4 minor 1 cited by

Weyl symmetry of the gradient-flow in information geometry

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper derives the information-geometric gradient flow from a Weyl-invariant action and identifies the flow with geodesic motion in a conformally rescaled Fisher metric.

desk verdict A correct but circular reformulation: the Weyl-invariant action encodes the gradient flow only because the flow is inserted through the momentum identification; the connection identities are the useful part. read the letter →

arxiv 2502.03866 v2 pith:HHNR3SCS submitted 2025-02-06 gr-qc cs.ITmath-phmath.ITmath.MP

Tatsuaki Wada , Sousuke Noda This is my paper · ORCID

classification gr-qccs.ITmath-phmath.ITmath.MP

keywords Weylintegrablegeometryinformationgradientflowalpha-connectionFishermetriceinbeinactionscaledgeodesic

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 tries to establish that the gradient-flow equations of information geometry are the Euler-Lagrange equations of a Weyl-invariant action, so that the flow is geodesic flow in a conformally rescaled Fisher metric. The central claim is that, in the Weyl gauge with gauge field $\omega_\mu = -\partial_\mu \ln \eta^2(\theta)$, the Weyl connection coincides with the Levi-Civita connection of the scaled metric $\tilde g_{\mu\nu} = \eta^2(\theta) g_{\mu\nu}$, and that this identity links the $\alpha$-connections at $\alpha = \pm 1$ to the Weyl-invariant connection. If the paper is right, the information-geometric gradient flow inherits a local scale symmetry, and the choice of gauge becomes part of the description of the flow. This matters because it supplies an action principle and a geometric picture for optimization dynamics in statistics, and it opens a concrete bridge between information geometry and Weyl-geometric gravity.

What carries the argument

The load-bearing object is the Weyl-invariant einbein action $S^a_{\rm IG} = \int \sqrt{\eta^2(\theta)}\,\sqrt{-ds^2}$, where $ds^2 = g_{\mu\nu}(\theta)d\theta^\mu d\theta^\nu$ is the extended $N+1$-dimensional line element built from the Fisher metric and the scalar $\eta^2(\theta)=g^{ij}\partial_i\Psi\,\partial_j\Psi$. The einbein field $e(\lambda)$ makes the action reparametrization-invariant, and the Euler-Lagrange equation for $e$ fixes $e = \sqrt{-ds^2}/\sqrt{\eta^2}$. Together with the gauge choice $\omega_\mu = -\partial_\mu\ln\eta^2$, this action converts the gradient-flow equations into autoparallel equations of the Weyl connection, and the identity ${}^w\Gamma = {}^{\tilde g}\Gamma$ shows that the same flow is geodesic in the conformally rescaled Fisher metric $\tilde g_{\mu\nu} = \eta^2 g_{\mu\nu}$.

What would settle it

Take a concrete exponential family, such as a one-dimensional Gaussian or Poisson model, and integrate the gradient-flow equation (15) numerically; construct $\eta^2(\theta)$ and the scaled metric $\tilde g = \eta^2 g$, then check whether the curve satisfies the geodesic equation of $\tilde g$ with proper-time parameter $\tau$ defined by $d\tau = \eta^2 dt$. If the geodesic equation fails for a generic initial condition, the claimed equivalence between the Weyl connection and the Levi-Civita connection of the scaled metric, as applied to gradient-flow trajectories, is false.

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Extended reading notes

Core claim

The paper's central claim is an equivalence, stated in Eqs. (59)--(61). With the positive scalar $\eta^2(\theta)=g^{ij}(\theta)\,\partial_i\Psi(\theta)\,\partial_j\Psi(\theta)$ and the Weyl gauge field $\omega_\mu = -\partial_\mu \ln \eta^2(\theta)$, the Weyl connection ${}^w\nabla$ is shown to equal the Levi-Civita connection ${}^{\tilde g}\nabla$ of the scaled metric $\tilde g_{\mu\nu} = \eta^2(\theta) g_{\mu\nu}$. Along the gradient-flow parameter $t$, its contraction with $\dot\theta$ reproduces the $\alpha=-1$ connection term: ${}^w\Gamma^i_{jk}\dot\theta^j\dot\theta^k = \Gamma^{(-1)\,i}_{jk}\dot\theta^j\dot\theta^k + (\partial_j\ln\eta^2\,\dot\theta^j)\dot\theta^i - \dot\theta^i$. The authors derive the gradient-flow equations (15) and their linearized form (18) from the Euler-Lagrange equations of a Weyl-invariant einbein action, and they relate the dual $\eta$-space version to the $\alpha=+1$ connection. In their reading, the information-geometric gradient flow is a time-like pre-geodesic flow in Weyl integrable geometry and an ordinary geodesic flow in the Einstein gauge of the scaled metric.

Load-bearing premise

The derivation assumes the static metric ansatz (36)-(37) and the identification of the information-geometric momentum $\eta_i$ with $\partial\Psi/\partial\theta^i$; if that identification is not accepted as an input, the Euler-Lagrange equations describe ordinary geodesic motion and the linear flow $d\eta_i/dt=\eta_i$ is not recovered.

Editorial extensions

If this is right

If correct, the gradient-flow equations (15) are the Euler-Lagrange equations of a Weyl-invariant action, so the flow carries a local scale symmetry rather than depending on one privileged coordinate choice.
The linearized flow $d\eta_i/dt = \eta_i$ in the dual coordinates is recovered from the same action, not assumed separately.
The $\alpha=-1$ (mixture) connection of information geometry appears as the $\theta$-space part of the Weyl autoparallel equations; the dual statement connects the $\alpha=+1$ connection to the $\eta$-space flow.
With the scaled metric $\tilde g_{\mu\nu} = \eta^2 g_{\mu\nu}$, the flow is ordinary geodesic flow, so standard geodesic techniques apply to gradient-flow trajectories.
The flow parameter $t$ is related to the Weyl proper time $\tau$ by $d\tau = \eta^2(\theta)\,dt$, giving the evolution parameter a geometric meaning.

Reading between the lines

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

An implicit consequence is that gauge freedom in this description is a design tool: changing $\eta^2$ by a conformal factor should amount to reparametrizing the same flow, which could be used to accelerate or regularize natural-gradient dynamics.
A direct numerical check is possible: for any exponential family, integrating (15) and testing whether the trajectory is a geodesic of $\eta^2 g$ would confirm or refute the geometric picture model by model.
If the action principle is taken literally, variational integrators built from it would preserve Weyl symmetry exactly and could yield structure-preserving algorithms for gradient-flow dynamics.
The paper demonstrates the connection only for $\alpha=\pm1$; extending it to intermediate $\alpha$ would require interpreting the family of $\alpha$-connections as a family of conformal gauges, a step the authors do not take.

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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

Worth knowing: the central claim in the abstract is too strong. The paper says the gradient-flow equations are derived from a Weyl-invariant einbein action, but that "derivation" only works because the flow is put in by hand through the correspondence p_i = eta_i = dPsi/dtheta^i. The Euler-Lagrange equations (43b) are second order; the gradient flow is first order. Without the identification p_i = eta_i, the action describes ordinary geodesic motion and the linear flow d eta_i/dt = eta_i is not recovered. So this is a reformulation, not a variational derivation.

What the paper does well: the algebra from the Lagrangian (39) through to (49) is internally consistent, and the conformal rescaling identity (59) is correct and clearly laid out. The identification of the Weyl connection with the Levi-Civita connection of the scaled metric eta^2 g, and the relation (61) to the alpha = -1 connection, are useful and do not appear verbatim in Ref. [15]. The paper also carefully checks the Weyl proper-time condition and the autoparallel/geodesic equivalence, which is a nice touch.

Soft spots, in proportion. The biggest is the circularity described above; that is fixable by dropping "derived" from the abstract and saying explicitly that the flow is an invariant submanifold selected by the phase-space dictionary. Two smaller issues. One, there is a sign inconsistency between the einbein Lagrangian (39) and the claimed action (50): (39) integrates to -integral sqrt(eta^2) sqrt(-ds^2), while (50) has the opposite sign. That needs a sign correction. Two, the authors do not state which equations are new relative to Ref. [15]; since the gauge field omega = -d ln eta^2 and the pre-geodesic form come from there, a clear new-vs-old list would help a reader judge the contribution.

The formal results (59)-(61) hold conditional on the gauge and the constrained flow, so the paper is a correct reformulation with a useful connection to Weyl geometry. With the overclaim removed and the sign fixed, it is publishable. I'd send it to a referee, with the request to check the variational claim and the sign.

Who this is for: people working on Weyl-geometric approaches to information geometry, or on alpha-connection relations in dually flat spaces. I would not cite it in my own work, but it is a serious paper.

Recommendation: accept for peer review, expect heavy-ish revision of the framing.

Referee Report

3 major / 4 minor

Summary. The paper proposes an (N+1)-dimensional, Weyl-invariant einbein action for the gradient-flow equations of information geometry. It introduces a static metric ansatz on the statistical manifold, identifies the canonical momentum with the expectation coordinate η_i = ∂Ψ/∂θ^i, and from Lagrangian (39) derives the gradient-flow equation (15) and its linearized form (49). It then shows that, in the Weyl gauge ω_μ = −∂_μ ln η²(θ), the Weyl connection coincides with the Levi-Civita connection of the conformally rescaled metric g̃ = η²g (Eq. (59)), and it relates the α = −1 connection to the Weyl connection (Eq. (61)). The central claim is that the gradient-flow equations are derived from a Weyl-symmetric action and that Amari's α-connections admit a Weyl-geometric interpretation.

Significance. If read as a constrained embedding rather than an unconstrained variational derivation, the paper gives a useful dictionary: the information-geometric gradient flow is realized as a geodesic/autoparallel flow in Weyl integrable geometry with a specific gauge, and the conformal-rescaling identity (59) is clean and standard. The computations are internally consistent and easy to verify, and the paper explicitly connects its formulas to the existing literature. However, the advertised statement that the gradient-flow equations are 'derived from the proposed action' is stronger than what is actually shown, because the phase-space identification p_i = η_i is assumed rather than derived. This does not invalidate the geometric relations, but it changes the nature of the contribution from a variational derivation to a constrained-sector equivalence.

major comments (3)

[§4.2, Eqs. (34b) and (43a)] The derivation of Eq. (15) from the Lagrangian (39) uses the correspondence p_i = η_i = ∂Ψ/∂θ^i as an input, not as a consequence of the action. From (39) with e(t)=1, the canonical momentum is p_i = g_{ij} dθ^j/dt; substituting p_i = η_i immediately reproduces (15). The same identification is fed into Eqs. (47)-(48) to obtain (49). The Euler-Lagrange equations (43b) are second order and admit generic solutions that do not satisfy p_i = η_i; the gradient flow is an invariant submanifold, not the general solution. Please either add a Lagrange-multiplier constraint enforcing p_i = ∂Ψ/∂θ^i, or explicitly state that the paper analyzes only the invariant sector p_i = η_i and adjust the abstract and conclusions accordingly.
[§4.2, Eq. (52) and Appendix B] The Weyl gauge field ω_μ = −∂_μ ln η²(θ) is imported from Ref. [15], and Appendix B derives it from the condition that the geodesic and autoparallel equations coincide for m² ↔ η². Thus the central equivalence (59)-(61) is conditional on a gauge choice that is an input convention rather than a prediction of the action. This should be stated explicitly; the current wording in the conclusions, 'we have shown, from the invariant action,' overstates what is actually derived.
[§4.2, Eqs. (36)-(38) and (42)] The derivation also relies on several additional inputs: the static metric ansatz (36), the proper-time relation (37) dτ = sqrt(−η² ds²), and the on-shell relation (38) that fixes g00 via (44). These choices are legitimate, but they are part of the model rather than consequences of the Weyl-invariant action. Please list them explicitly as assumptions in Section 4.2 so that the reader can distinguish the action-derived content from the gauge and parametrization choices.

minor comments (4)

[Eq. (16)] There is an index typo: the chain rule should read dθ^i/dt = (∂θ^i/∂η_j)(dη_j/dt), not dη_i/dt.
[Eqs. (36), (38), (44)] The symbol g00 is used for the covariant component in (36) but for the inverse component in (38) and (44), despite the note after (38). Please use g^{00} consistently for the inverse component.
[Reference [11]] The reference lists 'T. Noda' while the author list of the manuscript includes 'S. Noda'; please verify the correct initials.
[Eq. (55)] The step g_{ij}(θ) (dθ^i/dτ)(dθ^j/dτ) = 1/η²(θ) uses the relation dθ^i/dτ = (1/η²) dθ^i/dt, which is not stated at that point; adding one sentence would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

Gradient-flow 'derivation' relies on the momentum identification p_i = η_i, which is the flow itself; the conditional Weyl-connection identities retain independent content.

self definitional [Section 4.2, Eqs. (34b), (43a); cf. Eqs. (10) and (15)]
"First we make correspondence between the variables in analytical mechanics and those in IG as follows. ... momentum: p_i ↔ η_i, mass: m(x) ↔ √η^2(θ). ... η_i = g_{ij}(θ) dθ^j/dt , ... It is remarkable that the second equations in (43a) are equivalent to the gradient-flow equation (15)."

In information geometry η_i is defined by Eq. (10) as ∂Ψ/∂θ^i. Eq. (43a) sets the canonical momentum g_{ij}θdot^j equal to η_i, and with (10) this is exactly the gradient-flow equation (15), dθ^i/dt = g^{ij}∂Ψ/∂θ^j. The Euler-Lagrange equations (43b) are second order; they produce dP_i/dt = P_i for P_i = g_{ij}θdot^j. The first-order flow is obtained only on the invariant submanifold P_i = ∂Ψ/∂θ^i, which is not selected by the action but imposed by the dictionary (34b). Thus (15)/(18) are inputs renamed as canonical momenta, not independent outputs of the variational principle.

full rationale

The paper has a genuine non-circular core: the Weyl-connection identity (59) and the contraction identity (61) are algebraic consequences of the definitions once the gradient flow is assumed, and Eq. (59) is independent of the flow. The Weyl symmetry of the action (50) under (51) is also manifest. The circularity is confined to the claim that the gradient-flow equations (15)/(18) are derived from the action. The action's EL equations are second-order geodesic-type equations (53); to reduce them to the first-order flow, the paper imposes the phase-space correspondence (34b), p_i = η_i, and labels the canonical momentum as η_i in (43a). Since η_i is ∂Ψ/∂θ^i by definition, the asserted equivalence of (43a) with (15) is the gradient flow put in by hand, not a variational consequence. The same-author citation [15] supplies the Weyl gauge ω = -∂ ln η^2, but this is a convention rather than a load-bearing proof, and it does not by itself make (59) circular. The dP_i/dt = P_i equation and the scaled-metric identity show a real conditional equivalence, so the paper is not vacuous; however, the central 'derivation from first principles' reduces by construction to the input identification, giving partial circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the phase-space identification that already contains the gradient-flow equations and on a Weyl gauge field from Ref. [15], so the paper's derivation is largely a reformulation in a new geometric language.

assumptions (5)

standard math Exponential-family dually flat structure with Fisher metric g and alpha-connections (Eqs. 9-14).
Background from Amari-Nagaoka; assumed as the framework in which gradient flows are defined.
ad hoc to paper The metric of the N+1 spacetime is static and of the form ds^2 = g_{00}(theta) d tau^2 + g_{ij}(theta) d theta^i d theta^j, with d tau = sqrt(-eta^2 ds^2) (Eqs. 36-37).
This embedding ansatz is chosen specifically to make the particle analogy work; no independent physical motivation is given.
ad hoc to paper Canonical momentum is identified with eta_i = partial Psi/partial theta^i (Eq. 34b).
This identification already contains the gradient-flow equation (15); it is the load-bearing input that makes the Euler-Lagrange derivation reproduce the flow.
domain assumption On-shell mass-shell relation g^{mu nu} eta_mu eta_nu = -eta^2(theta) (Eq. 38), and the resulting g_{00} = -2/eta^2 (Eq. 44).
Transferred from the relativistic particle analogy p^2 = -m^2 extended to the information-geometric setting; it fixes the temporal metric component.
ad hoc to paper Weyl gauge field omega_mu = -partial_mu ln eta^2(theta) is adopted from Ref. [15].
The central gauge choice is not derived here; it is imported from the same author's prior work and is used to connect W-nabla to the scaled metric.

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Cite this review

Pith. "Pith review of Weyl symmetry of the gradient-flow in information geometry." pith.science (2026). https://pith.science/paper/HHNR3SCS

@misc{pith2026250203866,
  author       = {Pith},
  title        = {Pith review of: Weyl symmetry of the gradient-flow in information geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HHNR3SCS}},
  note         = {Machine review of arXiv:2502.03866}
}

abstract

We have revisited the gradient-flow in information geometry from the perspective of Weyl symmetry. The gradient-flow equations are derived from the proposed action which is invariant under the Weyl's gauge transformations. In Weyl integrable geometry, we have related Amari's $\alpha$-connections in IG to the Weyl invariant connection on the Riemannian manifold equipped with the scaled metric.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper

[15]

Wada, Weyl geometric approach to the gradient-flow equations in information geometry

T. Wada, Weyl geometric approach to the gradient-flow equations in information geometry. J. Geom. Symmetry Phys. , 66 59-70 (2023)

work page 2023
[1]

Weyl, Gravitation and electricity, Sitz

H. Weyl, Gravitation and electricity, Sitz. K¨ on. Preuss. Akad. Wiss. (1918) 465–480

work page 1918
[2]

and Salim J., Weyl Geometry and Gauge-Invariant Gravitation, Int

Poulis F. and Salim J., Weyl Geometry and Gauge-Invariant Gravitation, Int. J. Mod. Phys. D 23 (2014) 1450091

work page 2014
[3]

and Pucheu M., General Relativity and Weyl Geometry , Class

Romero C., Fonseca-neto J. and Pucheu M., General Relativity and Weyl Geometry , Class. Quantum. Grav. 29 (2012) 155015

work page 2012
[4]

Amari, H

S-I. Amari, H. Nagaoka, Methods of Information Geometry , American Mathematical Society, 2000

work page 2000
[5]

Amari, Information Geometry and Its Applications, Appl

S-I. Amari, Information Geometry and Its Applications, Appl. Math. Sci. 194, Tokyo, Springer 2016

work page 2016
[6]

Nielsen, An Elementary Introduction to Information Geometry, Entropy, 22, 1100

F. Nielsen, An Elementary Introduction to Information Geometry, Entropy, 22, 1100

work page
[7]

Caticha, The basics of information geometry , AIP Conf

A. Caticha, The basics of information geometry , AIP Conf. Proc. 1641, (2015), 15– 26

work page 2015

Show all 19 references

[8]

Ambrosio, N

L. Ambrosio, N. Gigli and G. Savarl´ e, Gradient Flows in Metric Spaces and in the Space of Probability Measures , 2nd Edn, Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel 2008

2008
[9]

Fujiwara and S-I

A. Fujiwara and S-I. Amari, Gradient Systems in View of Information Geometry , Physica D 80 (1995) 317–327

1995
[10]

Malag´ o and G

L. Malag´ o and G. Pistone, Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family, Entropy 17 (2015) 4215–4254. August 14, 2025 5:20 WSPC/INSTRUCTION FILE Weyl˙gauge˙symmetry˙in˙IG 16 T. Wada, S. Noda

2015
[11]

Boumuki and T

N. Boumuki and T. Noda, On Gradient and Hamiltonian Flows on Even Dimensional Dually Flat Spaces , Fundamental J. Math. and Math Sci. 6 (2016) 51–66

2016
[12]

and Matsuzoe H., An Eikonal Equation Approach to Thermo- dynamics and the Gradient Flows in Information Geometry , Physica A 570 (2021) 125820

Wada T., Scarfone A. and Matsuzoe H., An Eikonal Equation Approach to Thermo- dynamics and the Gradient Flows in Information Geometry , Physica A 570 (2021) 125820

2021
[13]

and Matsuzoe H., Huygens’ Equations and the Gradient-Flow Equations in Information Geometry, Int

Wada T., Scarfone A. and Matsuzoe H., Huygens’ Equations and the Gradient-Flow Equations in Information Geometry, Int. J. Geom. Methods Mod. Phys. 20, 2450012 (2023)

2023
[14]

Chanda and T

S. Chanda and T. Wada, Mechanics of geodesics in information geometry and black hole thermodynamics, Int. J. Geom. Methods Mod. Phys. 21, 2450098 (2024)

2024
[16]

Wada, A.M

T. Wada, A.M. Scarfone, A Hamiltonian approach to the gradient-flow equations in information geometry. Eur. Phys. J. B 97, 103 (2024)

2024
[17]

Avalos, F

R. Avalos, F. Dahia and C. Romero, A Note on the Problem of Proper Time in Weyl Space-Time, Found Phys 48, (2018) 253-270

2018
[18]

Kalashnikova, A.V

Yu.S. Kalashnikova, A.V. Nefediev, Relativistic quantum mechanics in the einbein field formalism , Yadernaya Fizika, 60, 1529–1535 (1997)

1997
[19]

Quiros, Gauge invariant theory of gravity in spacetime with gradient nonmetricity: A possible resolution of several cosmological puzzles , arXiv: 2301.05007

I. Quiros, Gauge invariant theory of gravity in spacetime with gradient nonmetricity: A possible resolution of several cosmological puzzles , arXiv: 2301.05007

arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [15]

Wada, Weyl geometric approach to the gradient-flow equations in information geometry

T. Wada, Weyl geometric approach to the gradient-flow equations in information geometry. J. Geom. Symmetry Phys. , 66 59-70 (2023)

work page 2023

[2] [1]

Weyl, Gravitation and electricity, Sitz

H. Weyl, Gravitation and electricity, Sitz. K¨ on. Preuss. Akad. Wiss. (1918) 465–480

work page 1918

[3] [2]

and Salim J., Weyl Geometry and Gauge-Invariant Gravitation, Int

Poulis F. and Salim J., Weyl Geometry and Gauge-Invariant Gravitation, Int. J. Mod. Phys. D 23 (2014) 1450091

work page 2014

[4] [3]

and Pucheu M., General Relativity and Weyl Geometry , Class

Romero C., Fonseca-neto J. and Pucheu M., General Relativity and Weyl Geometry , Class. Quantum. Grav. 29 (2012) 155015

work page 2012

[5] [4]

Amari, H

S-I. Amari, H. Nagaoka, Methods of Information Geometry , American Mathematical Society, 2000

work page 2000

[6] [5]

Amari, Information Geometry and Its Applications, Appl

S-I. Amari, Information Geometry and Its Applications, Appl. Math. Sci. 194, Tokyo, Springer 2016

work page 2016

[7] [6]

Nielsen, An Elementary Introduction to Information Geometry, Entropy, 22, 1100

F. Nielsen, An Elementary Introduction to Information Geometry, Entropy, 22, 1100

work page

[8] [7]

Caticha, The basics of information geometry , AIP Conf

A. Caticha, The basics of information geometry , AIP Conf. Proc. 1641, (2015), 15– 26

work page 2015