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

• URL: http://arxiv.org/abs/2511.08993v1
• Date: Thu, 13 Nov 2025 01:24:38 GMT
• Title: Fast $k$-means clustering in Riemannian manifolds via Fréchet maps: Applications to large-dimensional SPD matrices
• Authors: Ji Shi, Nicolas Charon, Andreas Mang, Demetrio Labate, Robert Azencott,
• Abstract summary: Key innovation is the use of the $p$-Fréchet map $Fp : mathcalM to mathbbRell$.<n>We rigorously analyze the mathematical properties of $Fp$ in the Euclidean space.<n>Our method reduces runtime by up to two orders of magnitude compared to intrinsic manifold-based approaches.
• Score: 4.958499383330019
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $F^p : \mathcal{M} \to \mathbb{R}^\ell$ -- defined on a generic metric space $\mathcal{M}$ -- which embeds the manifold data into a lower-dimensional Euclidean space $\mathbb{R}^\ell$ using a set of reference points $\{r_i\}_{i=1}^\ell$, $r_i \in \mathcal{M}$. 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 $\mathit{SPD}(n)$. Extensive numerical experiments using synthetic and real $\mathit{SPD}(n)$ 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.

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