Amari $m$-geodesic and dual flatness

Wiki card — concept node for exp-families-stability.

(i) Formal statement

Let $\mathcal{P}$ be a manifold of probability densities. Amari–Nagaoka define two dual affine connections on $\mathcal{P}$:

$e$-connection $\nabla^{(e)}$. Flat coordinates = natural (canonical) parameters $\theta$ of an exponential family. An $e$-geodesic is a curve $t \mapsto p_{\theta_0 + t \Delta\theta}$ — straight line in $\theta$.
$m$-connection $\nabla^{(m)}$. Flat coordinates = expectation parameters $\eta = \mathbb{E}_\theta[\varphi]$. An $m$-geodesic is a linear interpolation in density: $p_t = (1-t) p_0 + t p_1$ (mixture).

An exponential family $\mathcal{E}_\varphi = \{p_\theta \propto e^{-\theta^\top \varphi}\}$ is $e$-flat (canonical parameters are $e$-affine). A family is totally $m$-geodesic in $\mathcal{P}$ if for every pair $p_0, p_1 \in \mathcal{E}$, the $m$-geodesic between them stays in $\mathcal{E}$.

Pythagorean theorem. For $p \in \mathcal{E}_\varphi$ ($e$-flat), $q \in \mathcal{M}$ ($m$-flat), and $r = $ foot of $\nabla^{(e)}$-projection of $q$ onto $\mathcal{E}_\varphi$:

\[ \mathrm{KL}(q \| p) \;=\; \mathrm{KL}(q \| r) \;+\; \mathrm{KL}(r \| p). \tag{Pyth} \]

(ii) Role in Q1 / Q2 / Q3

Q1 (conceptual bridge). Stability of $\mathcal{E}_\varphi$ under convolution $p \mapsto p * \gamma_\sigma$ is a statement that the heat flow $p \mapsto P_t p = p * \gamma_{\sqrt{2t}}$ preserves $\mathcal{E}_\varphi$ as a submanifold of $\mathcal{P}$. In information-geometric language: $\mathcal{E}_\varphi$ is an invariant submanifold of the Fokker–Planck semigroup.
Q3 (HJ equivalence). On $\mathcal{E}_\varphi$, parameters $\theta_t$ evolve by an ODE (Riccati in Q2); this ODE is the restriction to $\mathcal{E}_\varphi$ of a vector field on $\mathcal{P}$ representing the heat flow. The Hamilton–Jacobi equation on $f = -\log p$ is the dual (via Legendre transform on the log-partition) of this ODE flow in $\eta$-coordinates.
Wheeler's invariant reformulation. The closure condition $(\dagger)$ is exactly the infinitesimal version of "$\mathcal{E}_\varphi$ is totally $m$-geodesic-stable under $P_t^*$": the Fokker–Planck vector field at $p_\theta$ decomposes into a tangential component in $\mathrm{Span}\{\varphi\}$ plus a normalisation (constant) component — nothing orthogonal, so the flow stays in $\mathcal{E}_\varphi$.

(iii) References

Amari (1985) — Differential-Geometrical Methods in Statistics, LNS 28. doi:10.1007/978-1-4612-5056-2.
Amari, Nagaoka (2000) — Methods of Information Geometry, AMS. §3.3 ($e$/$m$ connections), §3.5 (Pythagoras), §3.7 (e-families and submanifolds).
Amari (2016) — Information Geometry and Its Applications, Springer. doi:10.1007/978-4-431-55978-8.

(iv) Worked miniature — Gaussian family is $m$-geodesic-stable under convolution

Consider $\mathcal{E} = \{\mathcal{N}(\mu, \Sigma) : \Sigma \succ 0\}$ on $\mathbb{R}^d$, parametrised by $\varphi(x) = (\mathrm{vec}(xx^\top), x)$. The canonical $\theta$-parameter is $(K, h)$ with density $\propto \exp(-\tfrac12 x^\top K x + h^\top x)$; the expectation $\eta$-parameter is $(\mathbb{E}[xx^\top], \mathbb{E}[x]) = (\Sigma + \mu\mu^\top, \mu)$.

Closure under convolution. If $X \sim \mathcal{N}(\mu, \Sigma)$ and $Y \sim \mathcal{N}(0, \sigma^2 I)$ independent, then $X + Y \sim \mathcal{N}(\mu, \Sigma + \sigma^2 I)$. In $\theta$-coordinates $(K, h) = (\Sigma^{-1}, \Sigma^{-1}\mu)$:

\[ (K, h) \;\longmapsto\; \bigl((K^{-1} + \sigma^2 I)^{-1},\; (K^{-1} + \sigma^2 I)^{-1} K^{-1} h\bigr). \]

This is the Q2 reparameterisation $\theta \mapsto \tilde\theta$ in closed form. In the time-variable $t = \sigma^2/2$, infinitesimally $\dot K = -K^2$ (Riccati), $\dot h = -K h$ — the flow (3.5) of the invariant reformulation specialised to $(\mathbb{R}^d, \tfrac12\Delta)$.

Information-geometric read. The Fokker–Planck semigroup acts on $\mathcal{P}$, but when restricted to the $e$-flat submanifold $\mathcal{E}_{\mathrm{Gauss}}$ it remains in $\mathcal{E}_{\mathrm{Gauss}}$ — i.e. the Gaussian family is stable (invariant submanifold) under $P_t^*$. The flow is a curve in $\eta$-coordinates that is not a straight $m$-geodesic (it is nonlinear in $\eta$), but it stays in $\mathcal{E}_{\mathrm{Gauss}}$. This is the precise sense in which Q1 is a submanifold-invariance question, not a geodesic question.

Amari \(m\)-geodesic and dual flatness

(i) Formal statement

(ii) Role in Q1 / Q2 / Q3

(iii) References

(iv) Worked miniature — Gaussian family is \(m\)-geodesic-stable under convolution