Generalized infinite dimensional Alpha-Procrustes based geometries

TL;DR

This work develops a unified infinite-dimensional Alpha-Procrustes framework by extending unitized Hilbert–Schmidt operators and an extended Mahalanobis norm to define robust geometries for SPD operators. It shows that the infinite-dimensional GBW distance arises as a special case ($\alpha=\tfrac{1}{2}$) and the generalized Log-Hilbert–Schmidt distance emerges as $\alpha\to0$, with a regularization parameter $\rho$ ensuring stability and tractable spectrum handling. The authors provide both theoretical foundations and practical algorithms, including RGBW_∞ and spectrum-truncation-based Log-HS estimation, complemented by numerical experiments on diffusion-like operators and diffusion-geometry datasets. The framework supports spectrum truncation and learning of a data-adaptive metric, enabling robust, geometry-aware comparisons across spaces of varying dimension and scale, with potential applications in functional data analysis and graph-based learning. Overall, the paper offers a rigorous, adaptable pathway to extend Procrustes-type geometries to infinite dimensions, opening avenues for robust analysis of high- or kernel-based representations.

Abstract

This work extends the recently introduced Alpha-Procrustes family of Riemannian metrics for symmetric positive definite (SPD) matrices by incorporating generalized versions of the Bures-Wasserstein (GBW), Log-Euclidean, and Wasserstein distances. While the Alpha-Procrustes framework has unified many classical metrics in both finite- and infinite- dimensional settings, it previously lacked the structural components necessary to realize these generalized forms. We introduce a formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm that allows the construction of robust, infinite-dimensional generalizations of GBW and Log-Hilbert-Schmidt distances. Our approach also incorporates a learnable regularization parameter that enhances geometric stability in high-dimensional comparisons. Preliminary experiments reproducing benchmarks from the literature demonstrate the improved performance of our generalized metrics, particularly in scenarios involving comparisons between datasets of varying dimension and scale. This work lays a theoretical and computational foundation for advancing robust geometric methods in machine learning, statistical inference, and functional data analysis.

Figures (40)

• Figure 1: Comparison and evaluation of LES distance against IMD, GS and GW on 2D and 3D tori scaled by factor $c$ for $N=2000$ (number of eigenvalues estimated $= 200$). As $c \rightarrow 0$, $d(T_{2}, T_{2}^{Sc}), \, d(T_{3}, T_{3}^{Sc})$ and $d(T_{3}, T_{2}^{Sc})$ increases indicating the discrepancy between geometries of different dimensions. On the other hand $d(T_{2}, T_{3}^{Sc})$ since $\lim_{c\rightarrow0}T_{3}^{Sc} = T_{2}$shnitzer2022. As reported in the aforementioned paper, IMD_OURS with the LES descriptors is much less sensitive than LES with respect to the scaling in $c$ as indicated by the intersection of the lines for $d(T_{2}, T_{3}^{Sc})$ and $d(T_{2}, T_{2}^{Sc})$ for $c$ close to $1$.
• Figure 2: Comparison and evaluation of generalized LES distance against IMD, GS and GW on 2D and 3D tori scaled by factor $c$ for $N=2000$ and $\rho = 1.0 \times 10^{4}$ (number of eigenvalues estimated $= 200$). In the above we observe a similar trend as with LES in Figure \ref{['figLES']}. The distances with respect to GLES are overall lower and smoother as expected. At $N=2000$ we observe that the method seems less sensitive, the differences between tori are smaller, and there’s less contrast. However, the GLES performs much better for smaller values of $N$ and $\rho = 1.0 \times 10^{2}$ (see experiments in \ref{['generalisedLESDetailsResults']}).
• Figure 3: Comparison and evaluation of generalized LES distance with $\mathbf{\widetilde{M}}$ optimized against IMD, GS and GW on 2D and 3D tori scaled by factor $c$ for $N=2000$ and $\rho = 1.0 \times 10^{4}$ (number of eigenvalues estimated $= 200$). In the above we learn $\mathbf{\widetilde{M}}$ for cases where $K = 200, 500, 1000$. The distances with respect to an optimized GLES are lower and smoother (than LES) and also more sensitive to scaling with respect to different dimensions as seen is the separation of the lines for $d(T_{2}, T_{2}^{Sc})$ and $d(T_{3}, T_{3}^{Sc})$, which GLES (Figure \ref{['figGenLES']}) does not capture at large $N$.
• Figure 4: Comparing distance measures on 2D and 3D tori data with radii related by scaling c for $N = 200$ (first figure), $N=500$ (second figure), $N=1000$ and $N=2000$ (third and fourth figure) by reproducing the algorithm from shnitzer2022. Experiments become computationally expensive for sample sizes exceeding $N = 2000$.
• Figure 5: $N = 200$ and $\rho = 1.0$.

Theorems & Definitions (33)

• Definition 1.1
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Proposition 2.1
• Definition 3.1
• Proposition 3.1
• proof
• Corollary 3.1
• proof