Generalized infinite dimensional Alpha-Procrustes based geometries

Symmetric positive definite (SPD) matrices and operators are central to a wide range of problems in data science, including covariance estimation, kernel methods, diffusion geometry, and generative modeling. While the geometry of SPD matrices has been extensively studied in the finite-dimensional settingwith popular metrics such as the affine-invariant, Log-Euclidean, and Bures-Wasserstein (BW) distances, many real-world applications inherently involve infinite-dimensional SPD operators. These include covariance operators on functional spaces, integral kernels, and diffusion operators on manifolds. However, most existing geometric frameworks do not generalize coherently across finite and infinite dimensions, leading to inconsistencies in modeling, analysis, and computation. To address this, we propose a unifying family of Riemannian distances based on generalized alpha-Procrustes distances. This family includes the Log-Hilbert-Schmidt and infinite-dimensional GBW metrics as special cases and enables a continuous interpolation between them. Crucially, it is designed to extend smoothly from finite-dimensional SPD matrices to infinite-dimensional positive-definite Hilbert-Schmidt operators, offering a robust and flexible geometric foundation for both theoretical analysis and practical machine learning applications.