Alpha-Procrustes Riemannian Metrics

• Alpha-Procrustes metrics are a one-parameter family of Riemannian metrics on SPD matrices, operators, and Stiefel manifolds that interpolate between classical geometries.
• They admit explicit closed-form geodesics and distance formulas, enabling efficient and stable computation in both finite and infinite-dimensional settings.
• The framework unifies metrics such as log-Euclidean, Euclidean, and Bures-Wasserstein, providing robust tools for covariance comparison, functional data analysis, and statistical inference.

The Alpha-Procrustes family of Riemannian metrics constitutes a one-parameter continuum of metrics on manifolds of symmetric positive-definite (SPD) matrices, positive-definite operators, and Stiefel manifolds. These metrics interpolate between the log-Euclidean, Euclidean, Bures-Wasserstein, and other classical geometries. They admit explicit closed-form geodesics and distances and, in their infinite-dimensional generalization, provide robust, geometry-aware methods for covariance and operator comparison, functional data analysis, and statistical inference.

1. Formal Definitions and Mathematical Structure

The prototypical setting involves the manifold Sym⁺⁺(n) of $n\times n$ real symmetric positive-definite matrices. For $\alpha\in\mathbb{R}\setminus\{0\}$, define the finite-dimensional Alpha-Procrustes distance between $A,B\in$ Sym⁺⁺(n) as

$$d_\alpha(A,B) = \min_{U\in U(n)}\,\frac{1}{|\alpha|}\|A^\alpha - B^\alpha U\|_F$$

where $U^*$ is the unitary polar factor of $B^\alpha A^\alpha$. This distance can be computed in closed form: $$d_\alpha(A,B) = \frac{1}{|\alpha|}\,\mathrm{Tr}\left[A^{2\alpha}+B^{2\alpha}-2(A^\alpha B^{2\alpha} A^\alpha)^{1/2}\right]^{1/2}$$ (Quang, 2019, Thanwerdas et al., 2021).

In infinite dimensions, let $\mathcal{H}$ be a separable Hilbert space and $\Sigma_{\mathcal{S}_2^{(\mathrm{EXT})}}$ be the manifold of positive-definite extended Hilbert-Schmidt operators (unitization by $\mu I$). For $Y=X+\mu I$, define the extended Mahalanobis norm

$$\|Y\|_{M_\infty^{-1}} = \sqrt{\operatorname{tr}[(X+\mu I)^* (M+\rho I)^{-1}(X+\mu I)]}$$

with regularization $\rho > 0$ (Goomanee et al., 12 Nov 2025). The generalized Alpha-Procrustes distance is

$$d^\alpha_{ProM_\infty}(Y_1,Y_2) = \min_{U\in \mathcal{U}(\mathcal{H})}\ \|Y_1^\alpha - Y_2^\alpha U\|_{M_\infty^{-1}}$$

Special cases recover generalized Bures-Wasserstein at $\alpha=\frac{1}{2}$, log-Hilbert-Schmidt at $\alpha\rightarrow 0$.

On the Stiefel manifold $\mathrm{St}(n,p)$, the family is defined for tangent vectors $\xi = UA + U_\perp B$ by

$$g_\alpha(U)(\xi, \zeta) = (1-\alpha)\operatorname{tr}(\xi^T \zeta) + \alpha\operatorname{tr}\left(\xi^T (I - \frac{1}{2} U U^T)\zeta\right)$$

for $\alpha\in [0,1]$. This interpolates the Euclidean ($\alpha=0$) and canonical ($\alpha=1$) metrics (Mataigne et al., 18 Mar 2024, Zimmermann et al., 2021, Nguyen, 2020).

2. Geodesic Curves and Distance Formulas

Geodesics for SPD matrices admit explicit formulas: $$\gamma(t) = \left[ (1-t)^2 A^{2\alpha} + t^2 B^{2\alpha} + t(1-t)\big( (A^{2\alpha}B^{2\alpha})^{1/2} + (B^{2\alpha}A^{2\alpha})^{1/2}\big) \right]^{1/(2\alpha)}$$ with length $d_\alpha(A,B)$ (Quang, 2019, Thanwerdas et al., 2021).

On Stiefel manifolds,

$$\mathrm{Exp}^\alpha_U(\xi) = [U|Q]\exp\left(\begin{pmatrix} \beta A & B \ -B^T & 0 \end{pmatrix} \right)\begin{pmatrix} \exp(-(1-\alpha) A) \ 0 \end{pmatrix}$$

with $\beta=1-\alpha/2$ and $Q$ the orthogonal completion (Mataigne et al., 18 Mar 2024, Zimmermann et al., 2021). The corresponding logarithm map involves solving for $(A,B)$ in a nonlinear endpoint equation via matrix log and Sylvester equations.

In infinite dimensions, the geodesic for $Y_1,Y_2\in\Sigma_{\mathcal{S}_2^{(\mathrm{EXT})}}$ is realized as the alignment-minimizing curve in unitary orbit space, utilizing the polar decomposition (Goomanee et al., 12 Nov 2025).

3. Special Cases and Interpolating Limits

The Alpha-Procrustes family includes important limiting cases:

• $\alpha\to 0$: Log-Euclidean metric, $d_0(A,B)=\| \log A - \log B\|_F$
• $\alpha=1$: Euclidean metric, $d_1(A,B) = \|A-B\|_F$
• $\alpha=\frac{1}{2}$: Bures-Wasserstein/Bures distance, geometric mean, and Wasserstein-2 for Gaussian measures
• For the operator setting, $\gamma\to 0$ and $\alpha\geq \frac{1}{2}$ recovers trace-class Bures-Wasserstein; $\alpha\to 0$ gives log-Hilbert-Schmidt

The geodesics and metric tensors also interpolate between flat (zero-curvature) and nonnegative curvature regimes. Notably, mean-kernel conditions apply for $|\alpha|\leq\frac{1}{2}$, ensuring geodesic completeness (Thanwerdas et al., 2021).

4. Riemannian Metric, Curvature, and Geometric Properties

The finite-dimensional Alpha-Procrustes metric may be represented via the principle of deformed metrics: $$g^\alpha_\Sigma = \frac{1}{p^2} (\mathrm{pow}_p)^*g^{BW}$$ where $p=2\alpha$ and $\mathrm{pow}_p(\Sigma) = \Sigma^p$ (Thanwerdas et al., 2021). In joint eigenspace, this is a kernel metric with

$$\varphi_\alpha(x, y) = 2 x^{p-1} y^{p-1}(x^p + y^p)$$

Curvature analysis reveals that at the identity, all sectional curvature signs mirror those of the Bures-Wasserstein geometry. Along commuting directions, curvature is nonpositive in the infinite-dimensional version (Goomanee et al., 12 Nov 2025). The Alpha-Procrustes family generally lacks dually-flat structure, unlike mixed-Euclidean and $(\alpha,\beta)$-divergence metrics.

5. Infinite-Dimensional and Operator Extensions

The framework encompasses Hilbert-Schmidt positive-definite operators, with regularization ensuring norm boundedness and spectral stability. Core definitions invoke extended Mahalanobis norms and alignments over the unitary group (Goomanee et al., 12 Nov 2025, Quang, 2019). For RKHS covariance operators, all formulas reduce to kernel Gram matrix expressions, enabling direct calculation in kernelized and Gaussian process settings.

Distances between Gaussian measures with covariances $C_1, C_2$ are of the form: $$D_\alpha^2(\mathcal{N}(m_1,C_1), \mathcal{N}(m_2,C_2)) = \|m_1 - m_2\|^2 + \frac{1}{4\alpha^2} \mathrm{Tr}(C_1^{2\alpha} + C_2^{2\alpha} - 2(C_1^\alpha C_2^{2\alpha}C_1^\alpha)^{1/2})$$ with operator trace replacing finite trace as needed (Quang, 2019).

Regularization parameter $\rho$ and learnable Mahalanobis operator $M$ are essential to maintain numeric and geometric stability.

6. Computational Algorithms and Performance

Evaluation of Alpha-Procrustes metrics involves matrix powers, polar decompositions, and eigen-decompositions, typically at $O(n^3)$ cost per SPD matrix pair, or $O(m^3)$ for Gram-based RKHS computation (Quang, 2019, Goomanee et al., 12 Nov 2025). For infinite-dimensional operators, top-$K$ spectral truncation via randomized algorithms yields efficient approximations; error bounds for truncated norms are provided (Goomanee et al., 12 Nov 2025).

For Stiefel manifolds, the geodesic and logarithm maps are implemented in closed form using reduced $2p\times 2p$ block-matrix exponentials. Efficient algebraic algorithms for the logarithm map iterate over block structures with local linear convergence proven; per-iteration cost is $O(n p^2 + p^3)$, typically outperforming shooting methods when $p\ll n$ (Mataigne et al., 18 Mar 2024, Zimmermann et al., 2021).

Empirical comparisons indicate robust gains in numerical stability and discrimination power, especially for high-dimensional shape-comparison and Gaussian covariance benchmarking, when leveraging regularization and learnable metrics (Goomanee et al., 12 Nov 2025).

7. Connections to Broader Metric Families and Open Problems

The Alpha-Procrustes family fits within the general framework of deformed metrics via pull-back by matrix powers, and as such it is a one-parameter deformation of the Bures-Wasserstein geometry (Thanwerdas et al., 2021). Unlike mixed-Euclidean and $(\alpha,\beta)$-divergence families, it does not generically support a dually-flat metric structure.

Mean-kernel and kernel-metric representations distinguish metric completeness and positive-definiteness criteria, with explicit kernel characterization for $\alpha$ values in $|\alpha|\leq\frac{1}{2}$ (Thanwerdas et al., 2021). Notably, the family subsumes log-Euclidean, affine-invariant, and Wasserstein geometries as endpoints, and admits direct operator and kernel extensions for machine learning applications.

A plausible implication is that further parametrized generalization, regularization, and data-driven Mahalanobis selection will yield improved statistical and geometric robustness in high-dimensional functional data and geometric learning contexts.

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References:

• "Alpha Procrustes metrics between positive definite operators: a unifying formulation for the Bures-Wasserstein and Log-Euclidean/Log-Hilbert-Schmidt metrics" (Quang, 2019)
• "Generalized infinite dimensional Alpha-Procrustes based geometries" (Goomanee et al., 12 Nov 2025)
• "The geometry of mixed-Euclidean metrics on symmetric positive definite matrices" (Thanwerdas et al., 2021)
• "Computing the Riemannian logarithm on the Stiefel manifold: metrics, methods and performance" (Zimmermann et al., 2021)
• "Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics" (Nguyen, 2020)
• "An efficient algorithm for the Riemannian logarithm on the Stiefel manifold for a family of Riemannian metrics" (Mataigne et al., 18 Mar 2024)