Geometry Essentials

This page provides the minimal geometric background needed to understand why SPD Learn uses specialized operations. Full derivations and formulas live in Geometric Concepts, and we recommend the survey paper [Ju et al., 2026] for more details.

Why Geometry Matters

SPD matrices do not form a vector space: simple Euclidean operations (such as subtraction or arithmetic averaging) can break positive definiteness. The SPD manifold provides the appropriate setting for distances, averages, and interpolation.

Two ideas are used throughout the library:

• Tangent space: a local linearization that allows Euclidean tools.

• Log/Exp maps: conversions between SPD matrices and the tangent space.

Metric Choices in SPD Learn

A metric (specifically a Riemannian metric) defines how to measure distances and angles at each point on the manifold. It determines the “shortest path” (geodesic) between two SPD matrices and how they are averaged geometrically.

SPD Learn implements multiple metrics. The choice trades off invariance, stability, and computational cost.

Metric	When to use	Tradeoffs
AIRM	When affine invariance matters (e.g., within-session EEG)	Most faithful geometry, highest cost
LEM	Default for efficiency	Loses affine invariance
LCM	Large matrices or stability-first workflows	Less geometric fidelity than AIRM
BWM	When optimal-transport interpretation is useful	Different curvature; not affine invariant

In practice, Log-Euclidean (LEM) is a strong default. Use AIRM when affine invariance is a requirement rather than an optional benefit.

For details and formulas, see Geometric Concepts.