Local Fisher-Geometric Expansion

• Local Fisher-geometric expansion is a precise framework that defines local distances and curvature using the Fisher information metric in statistical manifolds.
• It contrasts with Euclidean and Wasserstein metrics by incorporating higher-order curvature corrections through cubic terms in parameter expansions.
• The approach extends to reaction-diffusion systems by applying geometric desingularization, yielding accurate asymptotic expansions for selected wave speeds.

The local Fisher-geometric expansion refers to a rigorous, explicit characterization of local distances, curvature, or wave speed expansions derived from the Fisher information geometry relevant to nonlinear dynamical systems and information manifolds. This concept admits two key mathematical realizations: first, in the finite-dimensional context of statistical families such as the multivariate normal, where the Fisher metric governs local distances; and second, in the geometric analysis of differential equations governing pushed reaction-diffusion fronts, where blow-up and desingularization produce local expansions of selected propagation speeds. Both settings systematically employ geometric, perturbative, and manifold-based techniques to obtain analytic expansions around singular or degenerate points.

1. Fisher Geometry: Metric Structure in Statistical Manifolds

The Fisher information metric equips families of probability distributions with a canonical Riemannian metric. For the $d$-variate normal family parameterized by mean $\mu \in \mathbb{R}^d$ and covariance $\Sigma \in \mathrm{Sym}^+_d$, the metric is given by

$$ds^2 = d\mu^T\, \Sigma^{-1} d\mu + \frac{1}{2} \mathrm{tr}\left[ (\Sigma^{-1} d\Sigma)^2 \right],$$

where the first term encodes Mahalanobis distances for mean shifts and the second penalizes covariance changes with a symmetrized Frobenius structure. The metric tensor $g_{IJ}(\theta)$ decomposes into block-diagonal form:

• $$g_{\mu_\alpha\mu_\beta} = (\Sigma^{-1})_{\alpha\beta}$$,
• $$g_{\Sigma_{ab}\Sigma_{cd}} = \frac{1}{2}\left[ (\Sigma^{-1})_{ac}(\Sigma^{-1})_{bd} + (\Sigma^{-1})_{ad}(\Sigma^{-1})_{bc} \right]$$, with all mixed blocks vanishing (Lawson et al., 2023).

2. Local Fisher–Geometric Expansion for Statistical Distances

The local expansion of the Fisher–Rao squared geodesic distance between parameter points $\theta$ and $\theta + \Delta\theta$ is formalized by

$$D_F^2(\theta, \theta + \Delta\theta) = g_{IJ}(\theta) \Delta\theta^I \Delta\theta^J + \frac{2}{3}\Gamma_{IJ,M}(\theta) \Delta\theta^I \Delta\theta^J \Delta\theta^M + \mathcal{O}(\|\Delta\theta\|^4),$$

where $\Gamma_{IJ,M} = g_{MK} \Gamma^K_{IJ}$ is the fully–lowered Christoffel symbol. The quadratic term is exact to second order: $$g_{IJ}\Delta\theta^I\Delta\theta^J = \Delta\mu^T \Sigma^{-1} \Delta\mu + \frac{1}{2} \mathrm{tr} [(\Sigma^{-1}\Delta\Sigma)^2],$$ representing combined Mahalanobis and covariance penalties. The cubic correction encodes intrinsic curvature, with the leading correction for the normal family given by

$$-\,\frac{4}{3}\, \Delta\mu^T \Sigma^{-1} \Delta\Sigma \Sigma^{-1} \Delta\mu - \frac{1}{3} \mathrm{tr} \bigl[(\Sigma^{-1}\Delta\Sigma)^3\bigr],$$

capturing mean–covariance coupling and nonlinear dependence of the Fisher geometry (Lawson et al., 2023).

3. Comparison with Euclidean and Wasserstein Metrics

The Fisher-geometric expansion can be contrasted with familiar alternatives:

• Euclidean: $\|\Delta\mu\|^2 + \|\Delta\Sigma\|^2_F$, ignoring covariance weighting and curvature,
• Wasserstein (2-Wasserstein): $\|\Delta\mu\|^2 + \|\Delta(\Sigma^{1/2})\|_F^2$, applying a different functional dependence to $\Delta\Sigma$. The Fisher expansion uniquely penalizes changes according to the local information geometry, naturally incorporating both Mahalanobis and symmetrized covariance structure. The cubic term is absent in both Euclidean and Wasserstein approaches, highlighting the Fisher geometry’s higher-order sensitivity to parameter changes.

Metric	Quadratic Penalty on $\Delta\mu$	Penalty on $\Delta\Sigma$	Curvature Corrections
Fisher	$\Delta\mu^T \Sigma^{-1}\Delta\mu$	$$\frac{1}{2}\mathrm{tr}[(\Sigma^{-1}\Delta\Sigma)^2]$$	Nonzero (cubic, etc.)
Euclidean	$\|\Delta\mu\|^2$	$\|\Delta\Sigma\|_F^2$	None
2-Wasserstein	$\|\Delta\mu\|^2$	$\|\Delta(\Sigma^{1/2})\|^2_F$	None

4. Geometric Desingularization and Front Propagation

In the nonlinear PDE context, the local Fisher-geometric expansion refers to the leading-order expansion for selected traveling wavespeeds in degenerate reaction–diffusion–advection systems. Examining the inviscid Fisher–KPP–Burgers system

$$\begin{cases} T_t = T_{xx} - (uT)_x + T(1-T),\ u_t = -u u_x + \rho T(1-T), \end{cases}$$

and passing to traveling coordinates with rescalings adapted to $\rho \to \infty$, the traveling-wave ODE system becomes singular at the steady state $(T,W,\epsilon) = (1, c, 0)$, where a loss of hyperbolicity precludes classical center manifold reductions. A quasi-homogeneous blow-up, employing chart decompositions with appropriately weighted polar coordinates, resolves this degeneracy, recovering hyperbolicity on the blown-up space. The stable and unstable manifolds in these charts can be computed explicitly, with the corresponding solution branches matched via a Shilnikov-type transition, ultimately leading to a nonlinear "mismatch" function whose root determines the corrected wave speed (Holzer et al., 2022).

5. Asymptotic Expansion of Selected Wavespeed

The local expansion for the selected wave speed follows from matching the strong stable ($g_s$) and unstable ($g_u$) manifold graphs at a fixed $T$: $$\Phi(c, \epsilon) = g_s(c, 1/2) - g_u(c, 1/2) + O(\epsilon).$$ The implicit function theorem guarantees a unique solution $c = c^*(\epsilon)$ near $\epsilon = 0$, yielding the expansion

$$c(\epsilon) = \left(\frac{3}{2}\right)^{1/3} + c_1 \epsilon + \mathcal{O}(\epsilon^2),$$

with $c_1$ defined via derivatives of $\Phi$ but not given in closed form. The main result for the original PDE parameters,

$$\tilde{c}^*(\rho) = \sqrt[3]{\frac{3\rho}{2} + O(1)}, \quad \rho \to \infty,$$

summarizes the local Fisher-geometric expansion of the wave speed near the singular limit (Holzer et al., 2022).

6. Validity, Higher-Order Terms, and Structural Consequences

All constructed manifolds and transition maps depend smoothly on the singular parameter ($\epsilon$), ensuring by geometric singular perturbation theory (Fenichel theory) that the wave speed expansion holds uniformly for small $\epsilon$. While higher-order terms could be computed via more refined matching or inertia-based continuation in the blown-up space, the leading $O(\epsilon)$ correction suffices to reveal structural consequences:

• The constructed front remains strictly monotonic, coincident with the strong-stable manifold at infinity,
• The front profile smoothly depends on problem parameters,
• The persistence and uniqueness of the pushed front is "pinned" transversely by the matching condition at the blown-up interface.

A plausible implication is that blow-up and local Fisher-geometric expansions provide a systematic analytic framework for understanding singular degeneration and selection mechanisms in high-dimensional dynamical and statistical systems (Holzer et al., 2022, Lawson et al., 2023).