Metric Correlation Structures in Riemannian Geometry

• Metric correlation structures are defined as frameworks that use Riemannian geometry and Lie group theory to construct intrinsic distances and statistical tools on full-rank correlation matrices.
• They address limitations of classical SPD metrics by employing quotient, Cholesky, and Hadamard metric constructions to ensure computational tractability and uniqueness of key descriptors.
• These structures enable closed-form geodesics and means, with practical applications in neuroimaging, signal processing, and high-dimensional data analysis.

Metric correlation structures refer to the class of geometric, algebraic, and statistical frameworks designed to rigorously paper correlation matrices and related objects within the context of metric geometry. These structures employ Riemannian geometry, Lie group theory, and matrix analysis to develop intrinsic distances, means, and statistical tools on the set of full-rank correlation matrices (the "open elliptope"). A core motivation is to overcome limitations of working in the wider space of SPD (symmetric positive definite) matrices, where classical metrics (e.g., the affine-invariant metric) do not naturally respect the normalization constraints or symmetries found in correlation matrices. Recent advances focus on defining metrics and geometric operations that not only account for these constraints but also ensure computational tractability, uniqueness of key statistical descriptors (like the Fréchet mean and the Riemannian logarithm), and flexibility for statistical inference and algorithms.

1. Quotient–Affine Metric on Full-Rank Correlation Matrices

The quotient–affine metric is constructed by descending the affine-invariant metric from the space of SPD matrices $\text{Sym}^+(n)$ to the space of full-rank correlation matrices $\text{Cor}^+(n)$. Given $\Sigma \in \text{Sym}^+(n)$, the affine-invariant metric is

$$g^{AI}_\Sigma(X, X) = \alpha\, \mathrm{tr}(\Sigma^{-1} X \Sigma^{-1} X) + \beta\,[\mathrm{tr}(\Sigma^{-1}X)]^2$$

with $\alpha > 0$ and $\beta > -\alpha/n$. Correlation matrices are obtained via

$$C = \mathrm{Diag}(\Sigma)^{-1/2}\; \Sigma\; \mathrm{Diag}(\Sigma)^{-1/2}$$

making $\text{Cor}^+(n)$ the quotient manifold $\text{Sym}^+(n)/\text{Diag}^+(n)$.

The quotient procedure involves splitting the tangent space at $\Sigma$ into vertical (tangent to $\text{Diag}^+(n)$-orbits) and horizontal (orthogonal complement) components. The induced metric at $C$ (up to scaling) is

$$g^{QA}_C(X,X) = \mathrm{tr}\left((C^{-1} X)^2\right) - 2\;\mathbf{1}^\top\,\mathrm{Diag}(C^{-1} X)\,[I_n + (C \circ C^{-1})]^{-1}\mathrm{Diag}(C^{-1} X)\,\mathbf{1}$$

where $\circ$ denotes the Hadamard product. This metric admits a closed-form exponential map, providing explicit geodesics, but crucially its sectional curvature is unbounded above, potentially precluding the uniqueness of the Riemannian logarithm and Fréchet mean (Thanwerdas et al., 2022).

2. Riemannian Operations and Lie–Cholesky Geometries

Standard Riemannian operations—exponential map, logarithm, parallel transport—are adapted to the quotient structure. The Cholesky decomposition enables a further reduction. Specifically, the group of lower triangular matrices with positive diagonal ($\mathrm{LT}^+(n)$) acts by conjugation. By pulling back the metrics via the Cholesky map, all geodesics and relevant Riemannian computations can be realized in the simpler manifold of lower-triangular matrices, often leading to numerically more stable algorithms (Thanwerdas et al., 2022).

Despite these computational benefits, the quotient–Lie–Cholesky metrics lack guaranteed upper curvature bounds and may still show non-uniqueness of means or logarithms, indicating the need for more regular structures.

3. New Families of Hadamard Metrics

To address these shortcomings, three novel families of metrics are introduced, all leveraging the Cholesky factorization:

Poly–Hyperbolic–Cholesky Metrics

• Each row $L_i$ of the Cholesky factor $L$ (subject to unit length) naturally lies on an open hemisphere of $S^{i-1}$.
• By a classical isometry, the open hemisphere is identified with a hyperbolic space $H^{i-1}$; thus, $\text{Cor}^+(n)$ is endowed with a product of hyperbolic metrics (“PHC metric”).
• This yields nonpositive (bounded below) sectional curvature—these are Hadamard manifolds, guaranteeing uniqueness of the logarithm and Fréchet mean.
• The PHC metric is defined by pulling back weighted product metrics from $H^1 \times \cdots \times H^{n-1}$ to $\text{Cor}^+(n)$ via the composite diffeomorphism $\Psi \circ \mathrm{Chol}$.

Euclidean–Cholesky Metrics

• By further flattening the geometry and pulling back the Euclidean metric from the Cholesky (lower triangular) domain, the metric becomes globally flat (curvature zero).
• All Riemannian operations (exponential, logarithm, distances, parallel transport) admit closed-form expressions.
• Uniqueness of the Fréchet mean is similarly ensured.
• In dimension 2, the Euclidean and log–Euclidean–Cholesky metrics coincide.

Log–Euclidean–Cholesky Metrics

• The Cholesky manifold is linearized by precomposing with the matrix logarithm, exploiting the nilpotency of strictly lower triangular matrices.
• This map gives a global diffeomorphism, ensuring computation of all Riemannian operations in closed form.
• The metric is again flat, and the uniqueness properties are preserved.

Crucially, all these metrics (especially Euclidean– and log–Euclidean–Cholesky) are not permutation-invariant, focusing on simplicity and uniqueness at the expense of full symmetry (Thanwerdas et al., 2022).

4. Nilpotent Lie Group Structure and Group Means

The space $\mathrm{LT}^1(n)$ of lower-triangular matrices with ones on the diagonal forms a nilpotent Lie group under matrix multiplication. The diffeomorphism

$\Theta:\;\mathrm{Cor}^+(n)\to\mathrm{LT}^1(n)$

transports the group structure onto $\mathrm{Cor}^+(n)$. The affine group exponential and logarithm maps are globally well-defined and unique due to nilpotency:

• Geodesics and means can be expressed entirely in terms of group operations.
• The group mean (Cartan–Schouten mean) is solved by

$\sum_{k=1}^N \log(\bar{\Gamma}^{-1}\Gamma_k) = 0$

for images $\Gamma_k = \Theta(C_k)$, then $\bar{G} = \Theta^{-1}(\bar{\Gamma})$.

• This provides a fully algebraic, closed-form approach for group averaging of correlation matrices, not available in general SPD settings.

5. Closed-Form Solutions and Dimension 2 Case

Many computations on $\text{Cor}^+(n)$ equipped with these novel metrics are available in closed form:

• For the Euclidean–Cholesky metric: the geodesic from $C$ to $C'$ is

$$\gamma(t) = \Theta^{-1}((1-t)\Theta(C) + t\Theta(C'))$$

• The geodesic distance is given by $\|\Theta(C')-\Theta(C)\|$.
• For $n=2$, the full-rank correlation matrix has the form $C(\rho)=\left[\begin{smaLLMatrix}1 & \rho \ \rho & 1\end{smaLLMatrix}\right]$, $\rho\in(-1,1)$:
  • Under the quotient–affine (or PHC) metric, geodesics follow

  $$\rho(t) = \frac{\rho_1 \cosh(\lambda t) + \sinh(\lambda t)}{\rho_1 \sinh(\lambda t) + \cosh(\lambda t)}$$

  for $$\lambda = \text{arctanh}(\rho_2) - \text{arctanh}(\rho_1)$$. - For the Euclidean/log–Euclidean–Cholesky metric, the geodesic is

  $$\rho(t) = \frac{F(t)}{\sqrt{1 + F(t)^2}},\quad F(t) = (1-t)\frac{\rho_1}{\sqrt{1-\rho_1^2}} + t\frac{\rho_2}{\sqrt{1-\rho_2^2}}$$ - In dimension two, the group mean and geodesic coincide for all new metrics (Thanwerdas et al., 2022).

6. Implications for Computation and Statistical Applications

• The Hadamard property (global nonpositive curvature) of the PHC, Euclidean–Cholesky, and log–Euclidean–Cholesky metrics guarantees global existence and uniqueness of Fréchet means and Riemannian logarithms, permitting robust computation of barycenters and means in large-scale applications.
• Closed-form solutions and the reduction to lower-triangular or nilpotent group operations ensure efficiency and numerical stability.
• Although permutation-invariance is absent in these structures, in various applications (e.g., in order-dependent data or regression settings) this is often a computational advantage.

7. Summary and Theoretical Advances

The introduction of new metric correlation structures—via Riemannian and Lie group techniques founded on the Cholesky factorization—solves longstanding issues in the geometry and statistics of correlation matrices:

• The quotient–affine metric offers a direct, intrinsic Riemannian structure but suffers from unbounded curvature and possible non-uniqueness of the mean.
• The three new families (poly–hyperbolic–, Euclidean–, and log–Euclidean–Cholesky metrics) endow the elliptope with Hadamard (or even flat) structure, ensuring uniqueness and computability of statistical operations.
• The nilpotent Lie group perspective yields unique group means defined algebraically.
• In dimension $n=2$, analytic geodesics and means can be written for all these metrics, illustrating the tractability and distinctions among constructions.

These advances provide a comprehensive geometric and statistical toolkit for the analysis, averaging, and inference on correlation matrices, with significant implications for fields such as statistical shape analysis, signal processing, neuroimaging, and high-dimensional data analysis (Thanwerdas et al., 2022).