amari-info-geom

Information geometry and statistical manifolds for the Amari library.

Overview

amari-info-geom implements the foundational concepts of information geometry, the study of probability distributions as points on a Riemannian manifold. This crate provides tools for working with Fisher metrics, α-connections, Bregman divergences, and the Amari-Chentsov tensor structure.

Named after Shun-ichi Amari, the pioneer of information geometry.

Features

• Fisher Information Metric: Riemannian metric on statistical manifolds
• α-Connections: Family of connections parameterized by α ∈ [-1, +1]
• Bregman Divergences: Information-theoretic divergence measures
• Statistical Manifolds: Geometric structures on probability spaces
• GPU Acceleration: Optional GPU support for large-scale computations
• Multivector Integration: Information geometry on Clifford algebra spaces

Installation

Add to your Cargo.toml:

[dependencies]
amari-info-geom = "0.12"

Feature Flags

[dependencies]
# Default features
amari-info-geom = "0.12"

# With GPU acceleration
amari-info-geom = { version = "0.12", features = ["gpu"] }

# High-precision arithmetic
amari-info-geom = { version = "0.12", features = ["high-precision"] }

Quick Start

use amari_info_geom::{kl_divergence, js_divergence, bregman_divergence};
use amari_core::Multivector;

type Cl3 = Multivector<3, 0, 0>;

// Create probability-like multivectors
let p = Cl3::from_coefficients(&[0.5, 0.3, 0.1, 0.1, 0.0, 0.0, 0.0, 0.0]);
let q = Cl3::from_coefficients(&[0.4, 0.4, 0.1, 0.1, 0.0, 0.0, 0.0, 0.0]);

// Compute divergences
let kl = kl_divergence(&p, &q)?;
let js = js_divergence(&p, &q)?;

// Bregman divergence with custom potential
let phi = |mv: &Cl3| mv.norm_squared();
let breg = bregman_divergence(phi, &p, &q)?;

Mathematical Background

Statistical Manifolds

A statistical manifold S is a set of probability distributions with a smooth structure:

S = { p_θ : θ ∈ Θ ⊂ ℝⁿ }

Fisher Information Metric

The Fisher information metric defines a Riemannian structure:

g_ij(θ) = E_θ[∂_i log p_θ · ∂_j log p_θ]

This metric measures the "distance" between nearby distributions.

α-Connections

A family of affine connections parameterized by α:

• α = +1: e-connection (exponential family)
• α = 0: Levi-Civita connection
• α = -1: m-connection (mixture family)

Dually Flat Manifolds

When α = +1 and α = -1 connections are both flat, the manifold is dually flat. This structure underlies:

• Exponential families
• Mixture families
• Bregman divergences

Divergences

Key Types

Parameter Trait

Objects usable as parameters on statistical manifolds:

use amari_info_geom::Parameter;

// Multivectors implement Parameter
let mv = Multivector::<3, 0, 0>::zero();
let dim = mv.dimension();
let component = mv.get_component(0);

GPU Operations

#[cfg(feature = "gpu")]
use amari_info_geom::{InfoGeomGpuOps, GpuStatisticalManifold};

let gpu_ops = InfoGeomGpuOps::new().await?;
let manifold = GpuStatisticalManifold::new(/* ... */);
let fisher = gpu_ops.compute_fisher_metric(&manifold).await?;

Modules

Applications

Machine Learning

• Natural Gradient Descent: Optimization respecting manifold geometry
• Variational Inference: KL minimization for approximate inference
• Wasserstein Distance: Optimal transport metrics

Statistics

• Model Selection: Comparing probability models
• Hypothesis Testing: Geometric interpretation of tests
• Sufficient Statistics: Exponential family structure

Physics

• Thermodynamics: Free energy and entropy
• Quantum Information: Quantum statistical manifolds
• Maximum Entropy: Inference with constraints

Theoretical Background

The crate is based on Amari's foundational work:

1. Dual Geometry: e-flat and m-flat coordinate systems
2. Pythagorean Theorem: D(p‖r) = D(p‖q) + D(q‖r) for orthogonal q
3. Amari-Chentsov Tensor: Unique invariant structure on statistical manifolds

References

• Amari, S. (2016). Information Geometry and Its Applications
• Amari, S., & Nagaoka, H. (2000). Methods of Information Geometry

License

Licensed under either of Apache License, Version 2.0 or MIT License at your option.

Part of Amari

This crate is part of the Amari mathematical computing library, named in honor of Shun-ichi Amari's contributions to information geometry.