Small-Variance Nonparametric Clustering on the Hypersphere

TL;DR

This paper introduces two novel k-means-like clustering algorithms for directional data on the hypersphere, leveraging small-variance limits of Bayesian nonparametric vMF mixtures, suitable for static and streaming data in 3D and high-dimensional applications.

Contribution

It proposes the DP-vMF-means and DDP-vMF-means algorithms, extending clustering to directional data with respect to the sphere's geometry, including a sequential version for streaming data.

Findings

Effective on synthetic and real 3D surface normals

Generalizes to high-dimensional directional data

Outperforms traditional clustering methods

Abstract

Structural regularities in man-made environments reflect in the distribution of their surface normals. Describing these surface normal distributions is important in many computer vision applications, such as scene understanding, plane segmentation, and regularization of 3D reconstructions. Based on the small-variance limit of Bayesian nonparametric von-Mises-Fisher (vMF) mixture distributions, we propose two new flexible and efficient k-means-like clustering algorithms for directional data such as surface normals. The first, DP-vMF-means, is a batch clustering algorithm derived from the Dirichlet process (DP) vMF mixture. Recognizing the sequential nature of data collection in many applications, we extend this algorithm to DDP-vMF-means, which infers temporally evolving cluster structure from streaming data. Both algorithms naturally respect the geometry of directional data, which lies on the unit sphere. We demonstrate their performance on synthetic directional data and real 3D surface normals from RGB-D sensors. While our experiments focus on 3D data, both algorithms generalize to high dimensional directional data such as protein backbone configurations and semantic word vectors.