Scalable Second-order Riemannian Optimization for $K$-means Clustering
Abstract
Clustering is a hard discrete optimization problem. Nonconvex approaches such as low-rank semidefinite programming (SDP) have recently demonstrated promising statistical and local algorithmic guarantees for cluster recovery. Due to the combinatorial structure of the -means clustering problem, current relaxation algorithms struggle to balance their constraint feasibility and objective optimality, presenting tremendous challenges in computing the second-order critical points with rigorous guarantees. In this paper, we provide a new formulation of the -means problem as a smooth unconstrained optimization over a submanifold and characterize its Riemannian structures to allow it to be solved using a second-order cubic-regularized Riemannian Newton algorithm. By factorizing the -means manifold into a product manifold, we show how each Newton subproblem can be solved in linear time. Our numerical experiments show that the proposed method converges significantly faster than the state-of-the-art first-order nonnegative low-rank factorization method, while achieving similarly optimal statistical accuracy.
Cite
@article{arxiv.2509.21675,
title = {Scalable Second-order Riemannian Optimization for $K$-means Clustering},
author = {Peng Xu and Chun-Ying Hou and Xiaohui Chen and Richard Y. Zhang},
journal= {arXiv preprint arXiv:2509.21675},
year = {2026}
}