A Riemannian Optimization Approach to Clustering Problems

Authors

Wen Huang*, Meng Wei, Kyle A. Gallivan, Paul Van Dooren

Abstract

This paper considers the optimization problem in the form of $\min_{X \in \mathcal{F}_v} f(x) + \lambda \|X\|_1,$ where $f$ is smooth, $\mathcal{F}_v = \{X \in \mathbb{R}^{n \times q} : X^T X = I_q, v \in \mathrm{span}(X)\}$, and $v$ is a given positive vector. The clustering models including but not limited to the models used by $k$-means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain $\mathcal{F}_v$ forms a compact embedded submanifold of $\mathbb{R}^{n \times q}$ and optimization-related tools are derived. An inexact accelerated Riemannian proximal gradient method is proposed and its global convergence is established. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method.

Key words

clustering, Riemannian optimization, proximal gradient method

Status

Submitted

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