Is Simple Uniform Sampling Efficient for Center-Based Clustering With Outliers: When and Why?

Clustering has many important applications in computer science, but real-world datasets often contain outliers. The presence of outliers can make the clustering problems to be much more challenging. In this paper, we propose a framework for solving three representative center-based clustering with outliers problems: $k$-center/median/means clustering with outliers. The framework actually is very simple, where we just need to take a small uniform sample from the input and run an existing approximation algorithm on the sample. However, our analysis is fundamentally different from the previous (uniform and non-uniform) sampling based ideas. To explain the effectiveness of uniform sampling in theory, we introduce a "significance" criterion and prove that the performance of our framework depends on the significance degree of the given instance. In particular, the sample size can be independent of the input data size $n$ and the dimensionality $d$, if we assume the given instance is sufficiently "significant", which is in fact a fairly appropriate assumption in practice. Due to its simplicity, the uniform sampling approach also enjoys several significant advantages over the non-uniform sampling approaches. The experiments suggest that our framework can achieve comparable clustering results with existing methods, but is much easier to implement and can greatly reduce the running times. To the best of our knowledge, this is the first work that systematically studies the effectiveness of uniform sampling from both theoretical and experimental aspects.