A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions (Free PDF)

 

A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions

Introduction

Probability distributions are central to Statistics, Data Science, Machine Learning, and Artificial Intelligence. A statistical model does not simply produce numbers—it describes uncertainty, variability, and relationships within data. This raises an interesting mathematical question:

How can we measure the distance between two probability distributions?

For ordinary points, we can use Euclidean distance. But probability distributions live in a much richer mathematical space. Their parameters can change simultaneously, their variances can change, and their underlying geometry is generally not flat.

This is where Information Geometry becomes important.

Frank Nielsen's research paper, “A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions,” presents a practical approach for approximating the Fisher–Rao distance between multivariate normal distributions. Published in Entropy in 2023, the paper addresses a particularly difficult problem: the Fisher–Rao distance between general multivariate normal distributions does not have a known closed-form expression.

The proposed approach approximates the distance by discretizing curves connecting two normal distributions and estimating the distances between neighboring distributions using the square root of their Jeffreys divergence. The paper also compares several parameterizations and a geometric construction based on the Calvo–Oller isometric embedding into a cone of symmetric positive-definite matrices.

Download the PDF for free: 

https://franknielsen.github.io/entropy-25-00654-v2.pdf

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What Is the Fisher–Rao Distance?

The Fisher–Rao distance comes from the Fisher information metric.

In information geometry, probability distributions are treated as points on a statistical manifold. The Fisher information provides a natural way to measure infinitesimal changes between nearby distributions.

The Fisher–Rao distance is then the length of the shortest geodesic connecting two distributions on this manifold.

In simple terms:

Fisher–Rao distance measures how statistically far apart two probability distributions are, while respecting the geometry of the statistical model.

This is fundamentally different from simply comparing their parameter values.

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Understanding Multivariate Normal Distributions

A multivariate normal distribution is described by two major components:

• Mean vector (\mu)

• Covariance matrix (\Sigma)

We can write it as:

[
N(\mu,\Sigma)
]

The mean controls the location of the distribution, while the covariance matrix determines its spread and correlations.

For a (d)-dimensional Gaussian, the parameter space contains:

• (d) mean parameters

• (\frac{d(d+1)}{2}) covariance parameters

Therefore, the dimension of the multivariate normal statistical manifold is:

[
\frac{d(d+3)}{2}
]

This rapidly becomes complicated as the dimensionality increases.

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Why Is the Problem Difficult?

For one-dimensional normal distributions, the Fisher–Rao distance has a closed-form expression.

However, for general multivariate normal distributions, a closed-form Fisher–Rao distance is not known.

This creates a computational challenge.

Researchers have investigated techniques such as:

• Geodesic shooting

• Numerical integration

• Upper bounds

• Lower bounds

• Geometric embeddings

But geodesic shooting can become computationally expensive and numerically unstable, particularly when the distributions are far apart. The paper specifically motivates its approximation method as a simpler alternative.

---

The Core Idea of the Paper

The main idea is surprisingly intuitive.

Suppose we have two normal distributions:

[
N_1
]

and

[
N_2
]

We construct a curve connecting them.

Instead of trying to calculate the exact geodesic directly, we divide the curve into many small segments.

Conceptually:

Distribution 1 → small step → small step → small step → Distribution 2

For sufficiently nearby distributions, their Fisher–Rao distance can be approximated using a divergence measure.

The individual small distances are then added together to approximate the total distance.

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Discretizing the Connecting Curve

Let a curve (c(t)) connect two distributions.

Instead of continuously calculating its Fisher length, we sample points:

[
c(0),c\left(\frac{1}{T}\right),c\left(\frac{2}{T}\right),\ldots,c(1)
]

This converts a continuous geometric problem into a sequence of smaller computational problems.

The approximation becomes:

Choose a curve → discretize it → calculate local distances → sum them.

This is the central computational idea behind the proposed method.

---

Jeffreys Divergence

The paper uses the square root of Jeffreys divergence to approximate the Fisher–Rao distance between nearby distributions.

Jeffreys divergence is the symmetrized version of KL divergence:

[
J(P,Q)=D_{KL}(P|Q)+D_{KL}(Q|P)
]

For sufficiently close distributions, this divergence provides useful local information about the Fisher geometry.

The approximation therefore avoids solving the complete Fisher–Rao geodesic problem directly.

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Why Use the Square Root?

The square root is important because divergence behaves locally like a squared distance.

For nearby distributions, the relationship between divergence and the Fisher metric allows a divergence-based quantity to act as an approximation to a local geometric distance.

This provides a computationally convenient way to estimate the length of each small segment.

---

Three Parameterizations

A particularly interesting part of the research is the comparison of different ways to represent multivariate normal distributions.

The paper investigates linear interpolation using:

Ordinary Parameters

The familiar representation:

[
(\mu,\Sigma)
]

Natural Parameters

Parameters associated with the exponential-family representation of the Gaussian distribution.

Expectation Parameters

Parameters based on expected sufficient statistics.

The choice of parameterization affects the resulting interpolation curve and therefore the quality of the approximation. The paper experimentally compares these alternatives.

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Why Parameterization Matters

Imagine two points in a geometric space.

A straight line between their coordinates looks simple.

But if we change the coordinate system, the same geometric space may no longer look straight.

The same phenomenon occurs with statistical distributions.

A linear interpolation in one parameterization does not necessarily correspond to a linear interpolation in another.

Therefore, choosing a suitable parameterization can significantly influence the quality of an approximation.

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Calvo–Oller Isometric Embedding

The paper also investigates a more geometric approach based on the Calvo–Oller isometric embedding.

The multivariate normal manifold can be embedded into the cone of symmetric positive-definite matrices of dimension ((d+1)\times(d+1)).

This is powerful because the complicated geometry of the Gaussian statistical manifold can be connected to a well-studied matrix geometry.

The paper compares the approximation based on interpolated curves with a curve derived from this embedding.

---

Symmetric Positive-Definite Matrices

A symmetric positive-definite (SPD) matrix satisfies:

[
x^\top A x>0
]

for every nonzero vector (x).

Covariance matrices are naturally SPD, which makes SPD geometry particularly relevant to multivariate statistics.

SPD matrices appear in:

• Covariance estimation

• Computer vision

• Signal processing

• Medical imaging

• Robotics

• Machine Learning

• Diffusion Tensor Imaging

The connection between Gaussian distributions and SPD matrices therefore has significant practical value.

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Fisher–Rao Geometry and Covariance Matrices

Covariance matrices do not merely contain numerical information about variance.

They also have geometric structure.

Two covariance matrices can differ in:

• Scale

• Orientation

• Correlation

• Eigenvalues

• Principal directions

Information geometry provides a principled framework for comparing these differences.

This becomes especially important when covariance matrices themselves are the primary objects of analysis.

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Special Cases With Exact Distances

Although the general multivariate problem does not have a known closed form, certain special cases do.

Same Mean

If two Gaussian distributions have the same mean but different covariance matrices, the Fisher–Rao distance can be expressed using generalized eigenvalues of the covariance matrices.

This makes covariance-only comparison mathematically tractable.

---

Same Covariance

If two Gaussian distributions have the same covariance but different means, the Fisher–Rao distance has a closed form involving the Mahalanobis distance.

The Mahalanobis distance measures separation relative to the covariance structure.

This is an important connection between classical multivariate statistics and information geometry.

---

Why Not Simply Use Euclidean Distance?

Suppose two Gaussian distributions have parameter vectors:

[
(\mu_1,\Sigma_1)
]

and

[
(\mu_2,\Sigma_2)
]

A simple approach would be to subtract their parameters and calculate Euclidean distance.

But this can be misleading.

Why?

Because:

• Covariance matrices are constrained objects.

• Different parameters have different statistical meanings.

• Reparameterization can change Euclidean distances.

• The geometry of probability distributions is not generally Euclidean.

Fisher–Rao geometry addresses these limitations by using the intrinsic geometry of the statistical model.

---

Applications in Diffusion Tensor Imaging

One particularly interesting application discussed in the paper is Diffusion Tensor Imaging (DTI).

DTI represents diffusion information using (3\times3) covariance-like matrices at locations throughout a three-dimensional grid.

These matrices can be associated with multivariate normal distributions.

This creates a large collection of Gaussian distributions whose pairwise distances may need to be calculated.

The paper notes that geodesic shooting can be expensive in this setting, making efficient approximations particularly useful.

---

Machine Learning Applications

The approximation method has potential relevance to Machine Learning problems involving probability distributions.

Possible applications include:

Distribution Clustering

Grouping similar Gaussian distributions.

Gaussian Mixture Models

Comparing components of probabilistic models.

Anomaly Detection

Identifying distributions that are statistically far from a reference model.

Probabilistic Embeddings

Representing uncertainty using Gaussian distributions.

Generative Models

Comparing learned probability distributions.

Time-Series Analysis

Measuring changes in local Gaussian statistics.

---

Information Geometry in Data Science

The paper demonstrates an important principle:

Data can have geometry.

When data is represented by probability distributions rather than individual points, ordinary distance metrics may no longer be sufficient.

Information geometry provides tools for working with:

• Probability distributions

• Statistical models

• Covariance matrices

• Divergences

• Geodesics

This makes it highly relevant to modern probabilistic Data Science.

---

Comparing the Main Approaches

The paper examines several approaches to the Fisher–Rao distance problem.

Approach	Main Idea	Advantage
Exact Fisher–Rao geodesic	Find the true shortest path	Mathematically ideal
Geodesic shooting	Numerically solve geodesic equations	General but computationally expensive
Curve discretization	Break a chosen curve into small segments	Simple and practical
Jeffreys-based approximation	Estimate local distances using divergence	Computationally convenient
Calvo–Oller embedding	Map Gaussian manifold into SPD geometry	Provides geometric structure

The proposed approximation is particularly attractive because it avoids the computational burden of directly solving the full geodesic problem.

---

Experimental Evaluation

The paper does not merely introduce the approximation—it evaluates its quality experimentally.

The author compares the numerical approximations against:

• Lower bounds

• Upper bounds

• Different interpolation strategies

• Calvo–Oller-based curves

This provides a way to assess how closely the proposed approximation approaches the true Fisher–Rao distance.

---

Computational Efficiency

One of the major motivations is computational practicality.

Exact or numerical geodesic calculations can become expensive, especially in high-dimensional settings.

The proposed approach instead relies on:

1. Selecting a tractable curve.

2. Discretizing that curve.

3. Calculating local divergences.

4. Summing the resulting local approximations.

This makes the method considerably easier to implement and experiment with.

---

The Bigger Picture

The importance of this paper extends beyond one distance measure.

It demonstrates how information geometry can turn difficult statistical problems into geometric problems.

The workflow is:

Probability distributions

↓

Statistical manifold

↓

Fisher information metric

↓

Geodesic distance

↓

Approximation through divergence

This connects probability theory, differential geometry, matrix geometry, and computational statistics.

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Skills You Can Develop

Studying this research can strengthen understanding of:

• Information Geometry

• Fisher Information

• Fisher–Rao Distance

• Multivariate Normal Distributions

• Probability Theory

• Statistical Manifolds

• Riemannian Geometry

• KL Divergence

• Jeffreys Divergence

• SPD Matrix Geometry

• Mahalanobis Distance

• Geodesics

• Numerical Approximation

• Machine Learning

• Statistical Computing

These concepts are especially valuable for advanced research in mathematical AI and probabilistic machine learning.

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Who Should Read This Paper?

This paper is particularly useful for:

Data Scientists

Interested in probability distributions and advanced statistical distances.

Machine Learning Researchers

Working with probabilistic models.

Statisticians

Exploring geometric approaches to multivariate distributions.

Mathematicians

Interested in Riemannian and information geometry.

AI Researchers

Studying geometry-aware learning methods.

Graduate Students

Looking for research topics connecting statistics and machine learning.

A background in probability, linear algebra, multivariate statistics, and basic differential geometry is helpful.

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Why This Paper Stands Out

Several aspects make this research particularly interesting:

• Tackles a difficult distance problem for multivariate Gaussians.

• Proposes a relatively simple approximation strategy.

• Uses Jeffreys divergence to approximate local Fisher–Rao distances.

• Compares ordinary, natural, and expectation parameterizations.

• Connects Gaussian geometry with SPD matrix geometry.

• Investigates the Calvo–Oller isometric embedding.

• Evaluates approximation quality against bounds.

• Connects theoretical information geometry with practical computation.

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Download the PDF for free: 

https://franknielsen.github.io/entropy-25-00654-v2.pdf

Conclusion

A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions provides an important contribution to computational Information Geometry. Frank Nielsen addresses a challenging problem: the Fisher–Rao distance between general multivariate normal distributions is not available in closed form, while direct numerical geodesic methods can be computationally demanding.

The proposed solution takes a practical route: construct a tractable curve between two Gaussian distributions, discretize it into smaller segments, approximate local Fisher–Rao distances using the square root of Jeffreys divergence, and combine those local estimates into an approximation of the overall distance. The research also compares different parameterizations and connects the Gaussian manifold to symmetric positive-definite matrix geometry through the Calvo–Oller embedding.

The paper is a valuable example of how geometry can make probability more understandable and computationally useful.

By connecting:

• Multivariate Normal Distributions

• Fisher Information

• Fisher–Rao Geometry

• Jeffreys Divergence

• KL Divergence

• Statistical Manifolds

• Geodesics

• SPD Matrices

• Mahalanobis Distance

• Information Geometry

• Numerical Approximation

• Machine Learning

the work provides a strong bridge between mathematical statistics and modern computational AI.

For researchers and advanced learners interested in Information Geometry, Probabilistic Machine Learning, Mathematical Statistics, or AI, this paper offers a fascinating look at how a difficult geometric distance can be approximated using elegant and computationally practical ideas.