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

2025-11-12

AI Summary

Clustering on high-dimensional non-Euclidean manifolds—such as the symmetric positive definite (SPD) matrix manifold—is computationally expensive and poorly scalable due to reliance on geodesic computations and iterative intrinsic optimization. To address this, we propose a fast $k$-means framework based on the $p$-Fréchet mapping. Our method constructs a reference point set to efficiently embed manifold-valued data into a low-dimensional Euclidean space, where standard $k$-means is then applied. Compared with conventional intrinsic manifold clustering algorithms, our approach bypasses costly geodesic calculations and iterative manifold optimization, achieving up to a 100× speedup while preserving clustering accuracy. It remains robust and stable under challenging conditions—including high dimensionality, large-scale datasets, and noise corruption. Our key contribution is the first systematic application of the $p$-Fréchet mapping for scalable, theoretically grounded embedding of manifold data into Euclidean space, establishing a new paradigm for efficient and principled non-Euclidean clustering.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $F^p : mathcal{M} o mathbb{R}^ell$ -- 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 $n imes n$ 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.

Problem

• Clustering high-dimensional data on non-Euclidean manifolds efficiently
• Overcoming computational challenges of intrinsic manifold clustering methods
• Enabling fast k-means clustering for large SPD matrices

Innovation

• Embedding manifold data into Euclidean space
• Using Fréchet maps for dimensionality reduction
• Applying k-means clustering after transformation