Uniform and Non-uniform Sampling Methods for Sub-linear Time k-means Clustering

Yuanhang Ren, Ye Du

Abstract: The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -means problem is arguably the most well-known clustering problem in machine learning, and lots of approximation algorithms have been proposed for it. However, many of these algorithms may become infeasible when data is huge. Sub-linear time algorithms with constant approximation ratios are desirable in this scenario. In this paper, we first improve the analysis of the algorithm proposed by [1] by sharpening the approximation ratio from 4( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</i> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</i> ) to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</i> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</i> . Moreover, on mild assumptions of the data, a constant approximation ratio can be achieved in poly-logarithmic time by the algorithm. Furthermore, we propose a novel sub-linear time clustering algorithm called <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Double</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-K-MC2</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sampling</i> as well. Experiments on the data clustering task and the image segmentation task have validated the effectiveness of our algorithms.