multi-scale-information-geometry-neural

name

multi-scale-information-geometry-neural

description

Multi-scale information geometry framework for analyzing neural population codes. Extends Fisher information metric across stimulus coarse-graining scales to reveal mutual information structure. Use when analyzing: (1) neural population coding geometry, (2) Fisher information limitations in neural data, (3) representational geometry from first principles, (4) mutual information estimation from neural responses, (5) diffusion model-based neural encoding analysis. Trigger: multi-scale Fisher, information geometry neural, representational geometry, neural population code, Fisher information metric, coarse-graining, mutual information neural.

Multi-Scale Information Geometry for Neural Populations

Based on Azeglio et al. (2026), arXiv:2605.06304.

Core Insight

The Fisher information metric is purely local — two neural codes can have identical Fisher information everywhere yet differ markedly in global discriminability. A unique multi-scale Riemannian geometry emerges from axioms about how distances contract under stimulus coarse-graining (isotropic Gaussian diffusion).

Key Mathematical Framework

Multi-Scale Fisher Information Matrix

G(x) = ∫₀^∞ [φ_t * J_t](x) dt

Where:

• φ_t = isotropic Gaussian kernel of variance t
• J_t(x) = Fisher information of the diffused conditional p_t(r|x)
• * = spatial convolution

Four Axioms Deriving the Geometry

1. Contraction: Coarse-graining reduces discriminability → distances contract
2. Locality: Contraction depends only on p(r|x) and first derivatives at x
3. Sufficiency: Invariant to sufficiency-preserving transformations of r
4. Zero baseline: If r ⊥ x, then G(x) = 0

Fundamental Identity: Mutual Information = Expected Magnification