Fast $k$-means clustering in Riemannian manifolds via Fréchet maps: Applications to large-dimensional SPD matrices

We introduce a novel, efficient framework for clustering data on high-dimensional, non-Euclidean manifolds that overcomes the computational challenges associated with standard intrinsic methods. The key innovation is the use of the $p$-Fréchet map $F^p:\mathcal{M}\to {\mathbb{R}}^{\mathrm{\ell }}$ – defined on a generic metric space $\mathcal{M}$ – which embeds the manifold data into a lower-dimensional Euclidean space ${\mathbb{R}}^{\mathrm{\ell }}$ using a set of reference points $\{r_i{\}}_{i=1}^{\mathrm{\ell }}$, $r_i\in \mathcal{M}$. Once embedded, we can efficiently and accurately apply standard Euclidean clustering techniques such as k-means. We rigorously analyze the mathematical properties of $F^p$ in the Euclidean space and the challenging manifold of $n\times n$ symmetric positive definite matrices $SPD(n)$. Extensive numerical experiments using synthetic and real $SPD(n)$ data demonstrate significant performance gains: our method reduces runtime by up to two orders of magnitude compared to intrinsic manifold-based approaches, all while maintaining high clustering accuracy, including scenarios where existing alternative methods struggle or fail.