fisherrao

Fisher-Rao geodesic distance on the multivariate Gaussian manifold — pure numpy + scipy. Production-grade implementation of the Calvo-Oller 1991 SPD embedding, closed-form, no C++ deps, no iterative solvers, no GPU.

What this computes

For two d-variate Gaussians N(μ_A, Σ_A) and N(μ_B, Σ_B), this library computes

d_CO(A, B) = || log( Q_A^{-1/2} Q_B Q_A^{-1/2} ) ||_F

where

Q(μ, Σ) = [[ Σ + μμᵀ, μ ], [ μᵀ, 1 ]] (the (d+1)x(d+1) Calvo-Oller SPD embedding)

This is a closed-form lower bound on the true Fisher-Rao geodesic distance between the two MVNs. It respects Chentsov's axioms (invariant under reparametrization), is symmetric and non-negative, vanishes iff the two MVNs are identical, and composes as a proper metric. For OMEGA-style applications (detecting regime shifts via velocity/acceleration of a rolling MVN trajectory) it is the right tool — what matters is a consistent, tractable Riemannian distance, not the exact Fisher-Rao value.

The true Fisher-Rao distance between arbitrary MVN pairs has no universally-accepted closed form; most production implementations (including this one) use either the Calvo-Oller lower bound or the Strapasson 2016 iterative solver. We ship the closed-form path because it's ~100× faster and plenty accurate for geodesic-velocity work.

Install

pip install fisherrao

Requires Python ≥ 3.9, numpy ≥ 1.23, scipy ≥ 1.10.

Quickstart

import numpy as np
from fisherrao import calvo_oller_distance, fit_mvn, trajectory_metrics

# two MVNs
mu_a = np.array([0.0, 0.0])
sigma_a = np.eye(2)
mu_b = np.array([1.0, 0.0])
sigma_b = np.array([[2.0, 0.3], [0.3, 1.0]])

d = calvo_oller_distance(mu_a, sigma_a, mu_b, sigma_b)   # => ~1.2

# fit an MVN from a sample matrix (shape n_samples x d) with automatic
# Ledoit-Wolf shrinkage — safe for small n
samples = np.random.default_rng(0).multivariate_normal(mu_a, sigma_a, size=40)
mu_hat, sigma_hat, shrinkage = fit_mvn(samples)

# trajectory velocity + acceleration from a sequence of MVNs
steps = trajectory_metrics(
    [(mu_a, sigma_a), (mu_b, sigma_b), (mu_b, 2 * sigma_b)],
    times=[0.0, 1.0, 2.0],
)
steps[1].velocity         # distance/time from step 0 to step 1
steps[2].acceleration     # velocity[2] - velocity[1], per unit time

Regime-shift detection (the motivating use case)

The examples/quickstart.py script generates 60 time steps of 10-dim samples with a mean shift at step 30, fits an MVN per step, and measures Fisher-Rao velocity between consecutive MVNs. On 50 samples per step, the velocity peaks exactly at step 30 at 2.4× the pre-shift baseline — a clean regime-detection signal.

Small-sample covariance

When the number of samples is smaller than the dimension, the empirical covariance is singular and Fisher-Rao distances blow up. fit_mvn defaults to Ledoit-Wolf 2004 shrinkage, a pure-numpy implementation matching scikit-learn's behaviour without the dependency. Shrinkage is returned alongside the fit so you can audit the regularisation strength.

API

Correctness

20 unit tests cover:

• SPD embedding positivity on randomised inputs
• d(A, A) = 0 up to 1e-9 in dimensions 1, 2, 5, 10
• Symmetry d(A, B) = d(B, A)
• Agreement between the fast eigenvalue path and the reference matrix-log path (1e-7 rtol)
• Pure mean-shift monotonicity in ||Δμ||
• Pure scale-shift closed form: d_CO(N(0, I_d), N(0, c·I_d)) = √d · |log c|
• Hand-derived 1D mean-shift formula: d_CO(N(0, 1), N(Δ, 1)) = √2 · acosh(1 + Δ²/2)
• Trajectory helpers: constant velocity ⇒ ~0 acceleration, strictly-increasing-times enforcement
• Ledoit-Wolf recovers ground-truth Σ on large n (< 15% relative Frobenius error) and keeps Σ SPD on n < d+1

Why this library exists

While building OMEGA's "Consensus Trajectory Geometry" meta-signal (Phase 19 Tier A #5 — Fisher-Rao geodesic velocity of sector-level agent-score MVNs as a pre-regime-transition indicator), we needed a small, fast, tested Fisher-Rao implementation that:

• Installs cleanly on Python 3.9–3.13 (geomstats 2.7 breaks on numpy 2.x as of 2026-04)
• Has zero C++ / GPU dependencies
• Documents the convention choice honestly (Calvo-Oller vs true Fisher-Rao is a real distinction)
• Validates against hand-derived analytical cases, not just library cross-checks

Nothing in the fisherrao API mentions OMEGA, regime shifts, or agent scores — the library is deliberately application-agnostic.

Roadmap

v0.2 (planned):

• Strapasson 2016 iterative solver for the true Fisher-Rao distance (when the lower bound's gap matters)
• Pinele-Costa 2020 sharper lower/upper bounds
• Fisher-Rao for Dirichlet / Beta / Gamma families (exponential family)
• Pre-whitening helper so trajectory distances ignore chosen axes' scales

Authors

• Pierre Samson (@darw007d) — idea, use-case, design decisions
• Claude Opus (Anthropic) — implementation and tests

Originally motivated by the OMEGA Swarm project, Phase 19 Tier A #5 "Consensus Trajectory Geometry". Shipped as a companion to phawkes (same authors, same "small, tested, publishable" ethos).

Citations

If this library contributes to a published result, please cite:

• Calvo, M. & Oller, J.M. (1991). A distance between multivariate normal distributions based in an embedding into the Siegel group. J. Multivariate Analysis 35(2).
• Ledoit, O. & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate Analysis 88(2).
• Pennec, X., Fillard, P. & Ayache, N. (2006). A Riemannian framework for tensor computing. Int. J. Computer Vision 66(1).
• Amari, S. (1985). Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics 28, Springer.

License

MIT — see LICENSE.