# Certified Sub-Linear Lower Bounds for K-Means Clustering

Soledad Villar (New York University)

Efficient algorithms for k-means clustering frequently converge to suboptimal partitions, and given a partition, it is difficult to detect k-means optimality. In this paper, we develop an a posteriori certifier of approximate optimality for k-means clustering. The certifier is a sub-linear Monte Carlo algorithm based on Peng and Wei's semidefinite relaxation of k-means. In particular, solving the relaxation for small random samples of the dataset produces a high-confidence lower bound on the k-means objective, and being sub-linear, our algorithm is faster than k-means++ when the number of data points is large. If the data points are drawn independently from any mixture of two Gaussians over R^m with identity covariance, then with probability 1?O(1/m), our poly(m)-time algorithm produces a 3-approximation certificate with 99% confidence (no separation required). We also introduce a linear-time Monte Carlo algorithm that produces an O(k) additive approximation lower bound.