Robust Algorithms for Online $k$-means Clustering

In the online version of the classic $k$-means clustering problem, the points of a dataset $u_1, u_2, …$ arrive one after another in an arbitrary order. When the algorithm sees a point, it should either add it to the set of centers, or let go of the point. Once added, a center cannot be removed. The goal is to end up with set of roughly $k$ centers, while competing in $k$-means objective value with the best set of $k$ centers in hindsight. Online versions of $k$-means and other clustering problem have received significant attention in the literature. The key idea in many algorithms is that of adaptive sampling: when a new point arrives, it is added to the set of centers with a probability that depends on the distance to the centers chosen so far. Our contributions are as follows: We give a modified adaptive sampling procedure that obtains a better approximation ratio (improving it from logarithmic to constant). Our main result is to show how to perform adaptive sampling when data has outliers ($\gg k$ points that are potentially arbitrarily far from the actual data, thus rendering distance-based sampling prone to picking the outliers). We also discuss lower bounds for $k$-means clustering in an online setting.