Notation

This page defines the mathematical notation used throughout SPD Learn documentation and source code. Consistent notation follows the conventions established in Congedo et al. [2017] and related literature.

Spaces and Manifolds

Symbol	Description	Example
\(\mathcal{S}^n_{++}\)	SPD manifold: \(n \times n\) symmetric positive definite matrices	[Congedo et al., 2017]
\(\text{Sym}(n)\)	Space of \(n \times n\) symmetric matrices	Tangent space at identity
\(\mathcal{S}^n\)	Space of symmetric matrices (same as \(\text{Sym}(n)\))	Log-Euclidean codomain
\(\mathbb{R}^{n \times n}\)	Space of \(n \times n\) real matrices	General matrix space
\(\mathbb{R}^n\)	\(n\)-dimensional Euclidean space	Vector space
\(\mathcal{M}\)	Generic Riemannian manifold	Abstract manifold notation
\(\mathcal{L}_+\)	Cholesky space: lower-triangular matrices with positive diagonal	Log-Cholesky metric

Groups

Symbol	Description	Usage
\(GL(n)\)	General linear group: invertible \(n \times n\) matrices	Affine invariance
\(\text{St}(n, k)\)	Stiefel manifold: \(n \times k\) matrices with orthonormal columns	BiMap weights

Tangent Spaces

Symbol	Description	Convention
\(T_P \mathcal{M}\)	Tangent space at point \(P\) on manifold \(\mathcal{M}\)	Uppercase \(P\)
\(T_P \mathcal{S}^n_{++}\)	Tangent space at \(P\) on SPD manifold	\(\cong \text{Sym}(n)\)

Note

We use uppercase letters (e.g., \(P\), \(Q\)) for points on the SPD manifold to distinguish them from lowercase scalars or indices. This convention is consistent with Pennec et al. [2006].

The tangent space \(T_P \mathcal{S}^n_{++}\) is isomorphic to \(\text{Sym}(n)\), denoted by \(\cong\) (not \(\equiv\)).

Maps and Operations

Exponential and Logarithmic Maps

Symbol	Description	Domain → Codomain
\(\text{Exp}_P\)	Riemannian exponential map at \(P\)	\(T_P\mathcal{M} \to \mathcal{M}\)
\(\text{Log}_P\)	Riemannian logarithmic map at \(P\)	\(\mathcal{M} \to T_P\mathcal{M}\)
\(\exp(S)\)	Matrix exponential	\(\text{Sym}(n) \to \mathcal{S}^n_{++}\)
\(\log(X)\)	Matrix logarithm	\(\mathcal{S}^n_{++} \to \text{Sym}(n)\)

Cholesky Operations

Symbol	Description	Notes
\(\log_{\text{chol}}\)	Log-Cholesky logarithm	Applies log to diagonal
\(\exp_{\text{chol}}\)	Log-Cholesky exponential	Applies exp to diagonal
\(\text{tril}(L, k)\)	Lower-triangular part with offset \(k\)	\(k=-1\) excludes diagonal

Riemannian Metrics

Inner Products

Symbol	Definition	Metric
\(g^{\text{AIRM}}_P(v, w)\)	\(\langle P^{-1/2} v P^{-1/2}, P^{-1/2} w P^{-1/2} \rangle_F\)	Affine-Invariant
\(g^{\text{LEM}}_P(v, w)\)	\(\langle D_P \log(v), D_P \log(w) \rangle_F\)	Log-Euclidean
\(g^{\text{BW}}_P(V, W)\)	\(\text{tr}(\mathcal{L}_P[V] W)\)	Bures-Wasserstein
\(g^{\text{LCM}}_P(v, w)\)	See Geometric Concepts	Log-Cholesky

Distance Functions

Symbol	Formula	Metric
\(d_{\text{AIRM}}(A, B)\)	\(\| \log(A^{-1/2} B A^{-1/2}) \|_F\)	Affine-Invariant
\(d_{\text{LEM}}(A, B)\)	\(\| \log(A) - \log(B) \|_F\)	Log-Euclidean
\(d_{\text{BW}}(A, B)\)	\(\sqrt{\text{tr}(A) + \text{tr}(B) - 2\text{tr}((A^{1/2} B A^{1/2})^{1/2})}\)	Bures-Wasserstein
\(d_{\text{LCM}}(A, B)\)	\(\| \text{logchol}(L_A) - \text{logchol}(L_B) \|_F\)	Log-Cholesky

Special Symbols

Symbol	Meaning	LaTeX
\(I\) or \(I_n\)	Identity matrix (of size \(n \times n\))	I or I_n
\(\cong\)	Isomorphic to	\cong
\(\langle \cdot, \cdot \rangle_F\)	Frobenius inner product	\langle \cdot, \cdot \rangle_F
\(\| \cdot \|_F\)	Frobenius norm	\| \cdot \|_F
\(\text{tr}(\cdot)\)	Matrix trace	\text{tr}(\cdot)
\(\text{diag}(\cdot)\)	Diagonal matrix or diagonal elements	\text{diag}(\cdot)
\(\odot\)	Log-Euclidean multiplication	\odot
\(\circledast\)	Log-Euclidean scalar multiplication	\circledast
\(d(\cdot, \cdot)\)	Generic distance function on the manifold	d(\cdot, \cdot)
\(G(P_1, \ldots, P_k)\)	Geometric (Riemannian) mean of SPD matrices	G(P_1, \ldots, P_k)

Log-Euclidean Group Operations

The Log-Euclidean framework defines group operations on SPD matrices:

\[A \odot B = \exp(\log(A) + \log(B))\]

\[t \circledast A = \exp(t \cdot \log(A)) = A^t\]

Layer Notation

Symbol	Operation	Module
\(\text{ReEig}(X)\)	\(U \max(\Lambda, \varepsilon) U^\top\)	ReEig
\(\text{LogEig}(X)\)	\(U \log(\Lambda) U^\top\)	LogEig
\(\text{ExpEig}(X)\)	\(U \exp(\Lambda) U^\top\)	ExpEig
\(\mathcal{G}\)	Fréchet mean (batch normalization)	SPDBatchNormMeanVar

Common Variables

Symbol	Description	Type
\(X, A, B, P, Q\)	SPD matrices	\(\in \mathcal{S}^n_{++}\)
\(S, V, W\)	Symmetric matrices (tangent vectors)	\(\in \text{Sym}(n)\)
\(U\)	Orthogonal matrix (eigenvectors)	\(U^\top U = I\)
\(\Lambda\)	Diagonal matrix (eigenvalues)	\(\lambda_i > 0\)
\(L\)	Lower-triangular Cholesky factor	\(\in \mathcal{L}_+\)
\(W\)	Stiefel matrix (BiMap weights)	\(\in \text{St}(n, k)\)
\(n\)	Matrix dimension	Positive integer
\(t\)	Geodesic parameter	\(t \in [0, 1]\)

Documentation Macros

SPD Learn provides LaTeX-style macros for consistent notation in documentation. These macros work in both HTML (via MathJax) and PDF (via LaTeX) output.

Usage in RST files:

The SPD manifold :math:`\spd` equipped with the AIRM metric...

The distance :math:`\dairm{A}{B}` between matrices...

The tangent space :math:`\tangent{P}` at point :math:`P`...

Available Macros:

Macro	Renders As	Description
\spd	\(\mathcal{S}^n_{++}\)	SPD manifold
\sym	\(\text{Sym}(n)\)	Symmetric matrices
\manifold	\(\mathcal{M}\)	Generic manifold
\choleskyspace	\(\mathcal{L}_+\)	Cholesky space
\tangent{P}	\(T_P \mathcal{M}\)	Tangent space at P
\tangentspd{P}	\(T_P \mathcal{S}^n_{++}\)	Tangent space on SPD at P
\Exp{P}	\(\text{Exp}_P\)	Riemannian exponential at P
\Log{P}	\(\text{Log}_P\)	Riemannian logarithm at P
\dairm{A}{B}	\(d_{\text{AIRM}}(A, B)\)	AIRM distance
\dlem{A}{B}	\(d_{\text{LEM}}(A, B)\)	Log-Euclidean distance
\dbw{A}{B}	\(d_{\text{BW}}(A, B)\)	Bures-Wasserstein distance
\dlcm{A}{B}	\(d_{\text{LCM}}(A, B)\)	Log-Cholesky distance
\gairm{P}	\(g^{\text{AIRM}}_P\)	AIRM inner product at P
\frob{X}	\(\| X \|_F\)	Frobenius norm
\frobinner{X}{Y}	\(\langle X, Y \rangle_F\)	Frobenius inner product
\tr	\(\text{tr}\)	Trace operator
\diag	\(\text{diag}\)	Diagonal operator
\reeig	\(\text{ReEig}\)	ReEig layer
\logeig	\(\text{LogEig}\)	LogEig layer
\frechet	\(\mathcal{G}\)	Fréchet mean
\geomean	\(G\)	Geometric mean
\I or \In	\(I\) or \(I_n\)	Identity matrix
\transpose	\(^\top\)	Transpose symbol

Note

Macros are defined in docs/source/conf.py under mathjax3_config (for HTML) and latex_elements["preamble"] (for PDF). When adding new notation, update both locations and this reference table.

References

[1] (1,2)

Marco Congedo, Alexandre Barachant, and Rajendra Bhatia. Riemannian geometry for eeg-based brain-computer interfaces; a primer and a review. Brain-Computer Interfaces, 4(3):155–174, 2017. doi:10.1080/2326263X.2017.1297192.

[2]

Xavier Pennec, Pierre Fillard, and Nicholas Ayache. A riemannian framework for tensor computing. International Journal of Computer Vision, 66(1):41–66, 2006. doi:10.1007/s11263-005-3222-z.

See also

• Geometric Concepts — Detailed explanations of geometric operations

• Glossary — Definitions of key terms

• API Reference — API reference with mathematical specifications