Amari Formulas Overview

• Amari formulas are a family of expressions in information geometry that define score functions, Fisher metrics, connections, torsion, and curvature in various coordinate systems.
• They reframe classical statistical models—such as the univariate Gaussian—in both standard (μ,σ) and moment-based (ξ) coordinates, highlighting changes in metric and connection properties.
• Beyond geometry, Amari formulas extend to neural-field and Hopfield–Amari models, offering gradient-flow frameworks and dynamic system formulations.

“Amari formulas” is not a single universally fixed expression but a family of formulas associated with work of Shun-ichi Amari and the later Amari–Čencov tradition. In the literature represented here, the term is used in at least three technically distinct senses: first, for the standard formulas of information geometry such as score identities, Fisher metrics, affine connections, torsion, and curvature; second, for the Amari–Čencov $\alpha$-connections and related divergence-induced constructions on statistical manifolds and spaces of densities; and third, for formulas attached to Amari neural field equations and Hopfield–Amari dynamics (Assandje et al., 10 Jul 2025). A central recent example is the explicit re-expression of classical information-geometric formulas for the univariate Gaussian family in the nonstandard coordinates $\xi=(\mu,\mu^2+\sigma^2)$, where the first two moments serve as coordinates (Assandje et al., 10 Jul 2025).

1. Terminological scope

In one precise contemporary usage, “Amari formulas” means the standard formulas of information geometry—score functions, Fisher metric, connection coefficients, torsion, and curvature—written in a chosen coordinate system on a statistical manifold (Assandje et al., 10 Jul 2025). The 2025 Gaussian-manifold paper is explicit on this point: its contribution is not a new abstract formalism, but the re-expression of these standard objects in a new “dual coordinate system” for the $2$-parameter univariate Gaussian family.

A broader information-geometric usage appears in work on Amari–Čencov $\alpha$-connections, where the relevant formulas are divergence-induced metric and connection formulas, duality relations, curvature expressions, and geodesic equations on spaces of densities or on homogeneous spaces of diffeomorphism groups (Lenells et al., 2012). In that setting, the central formulas are not tied to one statistical model, but to the geometry induced by an $\alpha$-divergence or by pullback from a linear or $L^p$ space.

A distinct usage occurs in neural-field and Hopfield-style dynamical systems. There, “Amari formulas” denotes the equations of the Amari neural field, its linearizations, power spectra, stochastic gradient-flow formulations, entropy-production formulas, or the anti-sign descent laws of Hopfield–Amari networks (Kuehn et al., 2018). This suggests that the phrase functions more as a domain marker than as a canonical theorem name: its meaning depends on whether the surrounding subject is information geometry, functional geometry of densities, neural-field theory, or quadratic-form optimization.

2. Gaussian statistical manifolds and coordinate-explicit formulas

For the univariate Gaussian family,

$$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$

the usual parameter coordinates are $\theta=(\mu,\sigma)$, and the log-density is

$$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$

The score functions are

$$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$

and the basic mean-zero identity is

$\xi=(\mu,\mu^2+\sigma^2)$0

(Assandje et al., 10 Jul 2025).

The Fisher metric is defined by

$\xi=(\mu,\mu^2+\sigma^2)$1

For the Gaussian family in $\xi=(\mu,\mu^2+\sigma^2)$2-coordinates, the metric is diagonal: $\xi=(\mu,\mu^2+\sigma^2)$3 so

$\xi=(\mu,\mu^2+\sigma^2)$4

The $\xi=(\mu,\mu^2+\sigma^2)$5-representation of the affine connection is

$\xi=(\mu,\mu^2+\sigma^2)$6

torsion is

$\xi=(\mu,\mu^2+\sigma^2)$7

and curvature is encoded by

$\xi=(\mu,\mu^2+\sigma^2)$8

(Assandje et al., 10 Jul 2025).

The paper’s distinctive coordinate choice is

$\xi=(\mu,\mu^2+\sigma^2)$9

Because $2$0 and $2$1, this is a moment-type parametrization. The Jacobian

$2$2

shows the map is locally a diffeomorphism, and the transformed basis is

$2$3

(Assandje et al., 10 Jul 2025).

In these coordinates, the $2$4-representation basis becomes

$2$5

$2$6

The Fisher metric acquires off-diagonal terms: $2$7 so

$2$8

The inverse metric is

$2$9

(Assandje et al., 10 Jul 2025).

The same pattern holds for the connection coefficients. Several coefficients that vanish or are simple in $\alpha$0-coordinates become nontrivial in the $\alpha$1-frame, for example

$\alpha$2

The paper then concludes from

$\alpha$3

that torsion in the $\alpha$4-coordinates is “not free,” i.e. nonzero. It also reports nonzero curvature components such as

$\alpha$5

and presents a nonzero scalar-curvature expression, though some printed formulas are typographically damaged in the manuscript (Assandje et al., 10 Jul 2025).

3. Divergence, duality, and $\alpha$6-connections

A more classical family of Amari formulas starts from a divergence $\alpha$7, from which one induces a metric and affine connection by

$\alpha$8

These are the standard Amari–Čencov formulas used on finite-dimensional manifolds of densities and in infinite-dimensional realizations via diffeomorphism quotients (Lenells et al., 2012).

On the homogeneous space

$\alpha$9

identified with smooth probability densities, the descended metric is the infinite-dimensional Fisher–Rao metric, and the family of $\alpha$0-divergences is given for $\alpha$1 by

$\alpha$2

with endpoint formulas

$\alpha$3

The induced connections satisfy the usual Amari picture: $\alpha$4 and $\alpha$5 are dual, $\alpha$6 is Levi-Civita, and $\alpha$7 are flat (Lenells et al., 2012).

In one dimension, the Christoffel map of the $\alpha$8-connection is

$\alpha$9

with the special cases

$L^p$0

The corresponding geodesic equation for the Eulerian velocity $L^p$1 is

$L^p$2

which specializes to the Hunter–Saxton equation at $L^p$3 and to other generalized Proudman–Johnson equations at $L^p$4 (Lenells et al., 2012).

A different but closely related formulation arises on spaces of densities through $L^p$5-geometry. For

$L^p$6

the Amari–Čencov connection on $L^p$7 has the explicit formula

$L^p$8

while on $L^p$9,

$$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$0

On $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$1, the geodesics of this $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$2-connection coincide with the geodesics of the $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$3-Fisher–Rao metric $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$4; on $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$5, the paper states that the geodesic equations coincide iff $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$6, equivalently $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$7 (Bauer et al., 2023).

4. Riemannian realization and metricity questions

Recent work recasts the Amari–Čencov $$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$8-connections in explicitly Riemannian terms. On

$$S=\left\{p_\theta(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\; ;\; \theta=(\mu,\sigma)\in \mathbb{R}\times \mathbb{R}_+^* \right\},$$9

the $\theta=(\mu,\sigma)$0-connection is

$\theta=(\mu,\sigma)$1

and there exist Riemannian metrics

$\theta=(\mu,\sigma)$2

whose Levi-Civita connections are exactly these $\theta=(\mu,\sigma)$3 (Bauer et al., 1 Aug 2025).

For $\theta=(\mu,\sigma)$4, $\theta=(\mu,\sigma)$5 is the classical Fisher–Rao metric. For general $\theta=(\mu,\sigma)$6, $\theta=(\mu,\sigma)$7 is typically not $\theta=(\mu,\sigma)$8-invariant, even though the connection is. The paper emphasizes the distinction: the connections $\theta=(\mu,\sigma)$9 are invariant, but the metrics $$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$0 generally depend explicitly on the chosen background density $$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$1 (Bauer et al., 1 Aug 2025).

On $$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$2, the same metricity phenomenon does not hold in full generality. The projected connection

$$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$3

is Levi-Civita for some Riemannian metric on $$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$4 iff

$$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$5

For $$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$6, it is non-metric (Bauer et al., 1 Aug 2025).

The curvature formula on $$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$7 is

$$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$8

Thus $$l(x,\theta)= -\log(\sqrt{2\pi}\sigma)-\frac{(x-\mu)^2}{2\sigma^2}.$$9 are flat, $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$0 has constant sectional curvature $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$1, and $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$2. This supplies one of the cleanest modern realizations of the classical Amari pattern: duality between $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$3, a distinguished Levi-Civita midpoint at $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$4, and flat endpoint geometries at $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$5 (Bauer et al., 1 Aug 2025).

The same paper also treats finite-dimensional statistical models. For a parametric model $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$6 with log-likelihood $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$7, the Fisher–Rao metric is

$$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$8

and the $$\partial_\mu l(x,\theta)=\frac{x-\mu}{\sigma^2}, \qquad \partial_\sigma l(x,\theta)= -\frac{1}{\sigma}+\frac{(x-\mu)^2}{\sigma^3},$$9-connection coefficients are

$\xi=(\mu,\mu^2+\sigma^2)$00

For two-dimensional exponential families with non-flat Fisher–Rao metric, the paper states that $\xi=(\mu,\mu^2+\sigma^2)$01 are the metric cases, whereas $\xi=(\mu,\mu^2+\sigma^2)$02 are non-metric (Bauer et al., 1 Aug 2025).

5. Amari formulas in neural-field and Hopfield–Amari models

Outside information geometry, the phrase also designates formulas attached to the Amari neural field equation. In its deterministic form on a bounded domain $\xi=(\mu,\mu^2+\sigma^2)$03,

$\xi=(\mu,\mu^2+\sigma^2)$04

and in stochastic form,

$\xi=(\mu,\mu^2+\sigma^2)$05

Under symmetry, continuity, and positive-definiteness assumptions on $\xi=(\mu,\mu^2+\sigma^2)$06, the deterministic equation becomes an exact gradient flow in the nonlocal Hilbert space

$\xi=(\mu,\mu^2+\sigma^2)$07

with energy

$\xi=(\mu,\mu^2+\sigma^2)$08

Its gradient identity is

$\xi=(\mu,\mu^2+\sigma^2)$09

so the exact gradient-flow formula is

$\xi=(\mu,\mu^2+\sigma^2)$10

and the stochastic gradient system is

$\xi=(\mu,\mu^2+\sigma^2)$11

(Kuehn et al., 2018).

In a different deterministic analysis of the Amari equation, linearization around a stationary homogeneous state $\xi=(\mu,\mu^2+\sigma^2)$12 yields

$\xi=(\mu,\mu^2+\sigma^2)$13

with Fourier-space response

$\xi=(\mu,\mu^2+\sigma^2)$14

The corresponding perturbation power spectrum is

$\xi=(\mu,\mu^2+\sigma^2)$15

and for an instantaneous localized stimulus one obtains a $\xi=(\mu,\mu^2+\sigma^2)$16 high-frequency tail at $\xi=(\mu,\mu^2+\sigma^2)$17 (Salasnich, 2015).

In the long-wavelength regime, the same analysis produces the diffusion approximation

$\xi=(\mu,\mu^2+\sigma^2)$18

and, after retaining quadratic terms,

$\xi=(\mu,\mu^2+\sigma^2)$19

(Salasnich, 2015).

The stochastic linearized Amari model gives yet another family of formulas. With drift kernel

$\xi=(\mu,\mu^2+\sigma^2)$20

noise covariance

$\xi=(\mu,\mu^2+\sigma^2)$21

and stationary covariance operator $\xi=(\mu,\mu^2+\sigma^2)$22, the entropy-production rate is

$\xi=(\mu,\mu^2+\sigma^2)$23

Equilibrium holds iff

$\xi=(\mu,\mu^2+\sigma^2)$24

equivalently

$\xi=(\mu,\mu^2+\sigma^2)$25

Under translational invariance, each Fourier mode satisfies

$\xi=(\mu,\mu^2+\sigma^2)$26

and its entropy-production contribution is

$\xi=(\mu,\mu^2+\sigma^2)$27

(Lucente et al., 18 Oct 2025).

A further dynamical use of the term appears in Hopfield–Amari networks, where minimization of quadratic or Hermitian forms on real or complex hypercubes is governed by anti-sign update laws. In the real case,

$\xi=(\mu,\mu^2+\sigma^2)$28

and any minimizer of $\xi=(\mu,\mu^2+\sigma^2)$29 over $\xi=(\mu,\mu^2+\sigma^2)$30 satisfies

$\xi=(\mu,\mu^2+\sigma^2)$31

where $\xi=(\mu,\mu^2+\sigma^2)$32 is the symmetric zero-diagonal matrix obtained from $\xi=(\mu,\mu^2+\sigma^2)$33. In the complex case, minimizers of $\xi=(\mu,\mu^2+\sigma^2)$34 over the complex hypercube satisfy

$\xi=(\mu,\mu^2+\sigma^2)$35

(Ramamurthy et al., 2012).

6. Generalized duality on gauge structures

An even broader abstraction of “Amari formulas” appears in the study of gauge structures on vector bundles. For a regular metric structure $\xi=(\mu,\mu^2+\sigma^2)$36 and a connection $\xi=(\mu,\mu^2+\sigma^2)$37, the generalized Amari transform $\xi=(\mu,\mu^2+\sigma^2)$38 is defined by

$\xi=(\mu,\mu^2+\sigma^2)$39

Equivalently,

$\xi=(\mu,\mu^2+\sigma^2)$40

This is the vector-bundle generalization of the standard dual-connection relation from information geometry (Boyom et al., 2017).

The central fixed-point criterion is

$\xi=(\mu,\mu^2+\sigma^2)$41

together with the involution property

$\xi=(\mu,\mu^2+\sigma^2)$42

The construction is gauge-covariant: $\xi=(\mu,\mu^2+\sigma^2)$43 These formulas elevate Amari duality from statistical manifolds to arbitrary finite-rank real vector bundles (Boyom et al., 2017).

The same framework introduces the first-order operator

$\xi=(\mu,\mu^2+\sigma^2)$44

and the PDE

$\xi=(\mu,\mu^2+\sigma^2)$45

When $\xi=(\mu,\mu^2+\sigma^2)$46, the existence of nontrivial solutions is equivalent to metricity of the gauge structure in the paper’s sense. The two index functions derived from this setup, the metric index $\xi=(\mu,\mu^2+\sigma^2)$47 and the gauge index $\xi=(\mu,\mu^2+\sigma^2)$48, vanish exactly in the “regularly special” cases singled out there (Boyom et al., 2017).

7. Conceptual significance and limitations

Across these literatures, the unifying role of “Amari formulas” is structural rather than terminological. In statistical geometry, they provide the coordinate machinery for passing from log-likelihood derivatives to intrinsic objects such as the Fisher metric, affine connections, torsion, curvature, and geodesics (Assandje et al., 10 Jul 2025). In density-space geometry, they encode the dualistic $\xi=(\mu,\mu^2+\sigma^2)$49-family and its metric, flat, and curvature properties, often through explicit linearizing embeddings into function spaces (Bauer et al., 2023). In neural-field theory and related dynamics, they designate the governing equations, their gradient-flow or spectral reductions, and their stochastic nonequilibrium structure (Kuehn et al., 2018).

The recent Gaussian-manifold study also clarifies an important limitation. Although it speaks of characterizing invariants such as “the dual potential function and the Fisher metric,” it does not actually provide explicit convex potentials $\xi=(\mu,\mu^2+\sigma^2)$50, $\xi=(\mu,\mu^2+\sigma^2)$51, nor Legendre duality relations such as

$\xi=(\mu,\mu^2+\sigma^2)$52

What it does provide explicitly are the metric, connection, torsion, and curvature formulas in the $\xi=(\mu,\mu^2+\sigma^2)$53-coordinates (Assandje et al., 10 Jul 2025).

Taken together, these works indicate that the most stable meaning of “Amari formulas” is the family of formulas that make a geometric or dynamical structure computable: divergence-to-metric formulas, dual-connection formulas, coordinate expressions for Fisher metrics and curvature, explicit $\xi=(\mu,\mu^2+\sigma^2)$54-connection formulas, and the governing equations of Amari neural systems. A plausible implication is that the phrase persists because it names a style of structure-preserving reformulation rather than a single formulaic object.