Vector-Valued Distance on SPD

• The paper introduces a vector-valued distance that fully characterizes the relative position between SPD matrices via eigen-log decomposition.
• The topic is defined as the study of SPD matrices viewed as a Riemannian manifold, where the vector-valued distance encodes multi-directional geometric displacements.
• The computation utilizes eigendecomposition under an affine-invariant Riemannian metric, ensuring isometric invariance and consistency with scalar distance norms.

The vector-valued distance (VVD) on the manifold of symmetric positive definite matrices, denoted SPDₙ, provides a complete Riemannian invariant that encodes the full relative position of two points in the space. Unlike scalar Riemannian distances, the VVD delivers a higher-fidelity geometrical descriptor that supports both theoretical analysis and practical applications in manifold learning, geometric representation, and information geometry.

1. Geometric Structure of SPDₙ

SPDₙ is the set

$$SPDₙ = \{ P \in \mathbb{R}^{n \times n} \mid P^\top = P,\;\;x^\top P x > 0 \;\forall\,x\neq0 \}$$

comprising all real, symmetric, positive definite $n \times n$ matrices. As a manifold, SPDₙ is smooth and of dimension $n(n+1)/2$, with tangent space at $P$ identified as $T_PSPDₙ \cong Sₙ$, where $Sₙ$ is the vector space of real symmetric matrices.

The manifold possesses a rich structure as a Riemannian symmetric space. The general linear group $GL(n)$ acts transitively by congruence: $\Phi_M(P) = M P M^\top, \quad M \in GL(n),$ preserving the Riemannian structure by isometries. This action partitions into “translations” by $M \in SPDₙ$ (no fixed point), “rotations” by $Q \in O(n)$ (fixing the identity matrix $n \times n$0), and the involutive “reflection” $n \times n$1.

The affine-invariant Riemannian metric is given by

$n \times n$2

for $n \times n$3. Isometries under $n \times n$4 manifest as

$n \times n$5

2. Construction of the Vector-Valued Distance

For $n \times n$6, the VVD produces a vector in $n \times n$7 that characterizes the displacement from $n \times n$8 to $n \times n$9. The construction proceeds as follows:

• Compute $n(n+1)/2$0, which belongs to $n(n+1)/2$1 and is thus diagonalisable.
• Perform an orthogonal eigendecomposition:

$n(n+1)/2$2

where $n(n+1)/2$3, $n(n+1)/2$4.

• The Riemannian logarithm at the identity coincides with the matrix logarithm:

$n(n+1)/2$5

• By definition (Lopez et al. §2.2), the vector-valued distance is

$n(n+1)/2$6

with ordering $n(n+1)/2$7 ensuring placement in the “positive Weyl chamber.”

Each component $n(n+1)/2$8 measures the signed expansion along the $n(n+1)/2$9-th eigendirection, offering directional decomposition of displacement within the non-Euclidean geometry.

3. Properties of the Vector-Valued Distance

The VVD encapsulates several distinguished geometric and algebraic properties:

• Isometric Invariance: For any isometry $P$0 mapping $P$1 to $P$2, $P$3 and $P$4 share the spectrum, ensuring $P$5. Conversely, equal VVD vectors imply the pairs are isometric.
• Anti-symmetry (up to permutation): The displacement reverses sign (up to order reversal): if the eigenvalues of $P$6 are $P$7, then those of $P$8 are $P$9, yielding

$T_PSPDₙ \cong Sₙ$0

after reordering.

• Triangle (Majorization) Property: For all $T_PSPDₙ \cong Sₙ$1,

$T_PSPDₙ \cong Sₙ$2

with $T_PSPDₙ \cong Sₙ$3 the majorization order on $T_PSPDₙ \cong Sₙ$4. This property subsumes the ordinary triangle inequality under any symmetric norm.

• Complete Invariant: The VVD provides a complete isometry-invariant between pairs in $T_PSPDₙ \cong Sₙ$5; no geometric information is lost.

4. Connection to Scalar Distance and Norms

The classical affine-invariant Riemannian distance emerges as the $T_PSPDₙ \cong Sₙ$6-norm of the VVD: $T_PSPDₙ \cong Sₙ$7 More generally, all $T_PSPDₙ \cong Sₙ$8-norms of $T_PSPDₙ \cong Sₙ$9, invariant under coordinate permutations, yield Finsler metrics on $Sₙ$0 with the same group of isometries. Examples include: \begin{align*} d_{F_1}(P, Q) & = \sum_{i=1}^n |\log \lambda_i|, \ d_{F_\infty}(P, Q) & = \max_{i} |\log \lambda_i|. \end{align*} This structure generalizes the interpretation of distance beyond scalar length to encompass multi-directional geometric information.

5. Extension to General Symmetric Spaces

The vector-valued distance admits generalization to all noncompact Riemannian symmetric spaces $Sₙ$1 of rank $Sₙ$2: $Sₙ$3 defined by selecting a maximal flat through a reference point, conjugating an argument into it, and reading “log-coordinates.” SPDₙ forms a canonical instance with $Sₙ$4. Analogous constructions apply to Grassmannians, orthogonal groups, and beyond, rooting in the general Lie-theoretic setting of Weyl chambers and Cartan subalgebras (see Helgason (1978); Kapovich–Leeb–Porti (2017)).

6. Algorithmic Computation of VVD

The computation of the VVD between $Sₙ$5 proceeds via the following pseudocode:

1. Compute $Sₙ$6 (this may be solved as $Sₙ$7).
2. Perform eigendecomposition:

$Sₙ$8

1. Assemble $Sₙ$9 by $GL(n)$0 for $GL(n)$1.
2. Return $GL(n)$2.

The dominant computational cost arises from the eigendecomposition, which is $GL(n)$3. Implementation details, proofs, and further algorithmic considerations are discussed in the appendices of Lopez et al. (2021).

7. Applications and Visualization

The VVD supports a spectrum of applications in geometric representation learning. In tasks such as knowledge graph completion, item recommendation, and question answering, models utilizing SPDₙ and its VVD-based geometry demonstrate improved performance relative to analogous Euclidean and hyperbolic architectures. The vector-valued structure of the VVD also enables direct visualization of learned embeddings, revealing clear separation between positive and negative samples.

A plausible implication is enhanced interpretability: the decomposition of distances along Weyl chamber directions provides insights into how complex data relationships unfold in high-rank, non-Euclidean spaces.

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For a rigorous treatment of the geometric and algebraic properties, as well as applications and proof details, see the full development and appendices in "Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices" (López et al., 2021).