Spherical Principal Component Analysis

TL;DR

This paper introduces a novel spherical PCA method that incorporates angle distance constraints, providing a more suitable analysis tool for fields where angular relationships are critical, with proven convergence and superior clustering performance.

Contribution

It proposes a new spherical PCA approach with constraints on factors to unify Euclidean and angle distances, addressing nonconvex optimization challenges with an effective algorithm.

Findings

Validated on synthetic and real data, showing improved clustering results.

Demonstrated convergence and effectiveness of the proposed optimization method.

Abstract

Principal Component Analysis (PCA) is one of the most important methods to handle high dimensional data. However, most of the studies on PCA aim to minimize the loss after projection, which usually measures the Euclidean distance, though in some fields, angle distance is known to be more important and critical for analysis. In this paper, we propose a method by adding constraints on factors to unify the Euclidean distance and angle distance. However, due to the nonconvexity of the objective and constraints, the optimized solution is not easy to obtain. We propose an alternating linearized minimization method to solve it with provable convergence rate and guarantee. Experiments on synthetic data and real-world datasets have validated the effectiveness of our method and demonstrated its advantages over state-of-art clustering methods.