Fast $k$-means clustering in Riemannian manifolds via Fr\'{e}chet maps: Applications to large-dimensional SPD matrices

TL;DR

This paper presents a fast and efficient k-means clustering framework for high-dimensional non-Euclidean manifolds, especially SPD matrices, using Fréchet maps to embed manifold data into Euclidean space for scalable clustering.

Contribution

The authors introduce the $p$-Fréchet map for embedding manifold data into Euclidean space, enabling fast k-means clustering on high-dimensional non-Euclidean data, with rigorous analysis and extensive experiments.

Findings

Reduces runtime by up to two orders of magnitude compared to intrinsic methods.

Maintains high clustering accuracy on synthetic and real SPD data.

Effective even where existing methods struggle or fail.

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 $p$-Fr\'{e}chet 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.