On Sampling Based Algorithms for k-Means

We generalise the results of Bhattacharya et al. [Bhattacharya et al., 2018] for the list-k-means problem defined as - for a (unknown) partition X₁, ..., X_k of the dataset X ⊆ ℝ^d, find a list of k-center-sets (each element in the list is a set of k centers) such that at least one of k-center-sets {c₁, ..., c_k} in the list gives an (1+ε)-approximation with respect to the cost function min_{permutation π} [∑_{i = 1}^{k} ∑_{x ∈ X_i} ||x - c_{π(i)}||²]. The list-k-means problem is important for the constrained k-means problem since algorithms for the former can be converted to {PTAS} for various versions of the latter. The algorithm for the list-k-means problem by Bhattacharya et al. is a D²-sampling based algorithm that runs in k iterations. Making use of a constant factor solution for the (classical or unconstrained) k-means problem, we generalise the algorithm of Bhattacharya et al. in two ways - (i) for any fixed set X_{j₁}, ..., X_{j_t} of t ≤ k clusters, the algorithm produces a list of (k/(ε))^{O(t/(ε))} t-center sets such that (w.h.p.) at least one of them is good for X_{j₁}, ..., X_{j_t}, and (ii) the algorithm runs in a single iteration. Following are the consequences of our generalisations: 1) Faster PTAS under stability and a parameterised reduction: Property (i) of our generalisation is useful in scenarios where finding good centers becomes easier once good centers for a few "bad" clusters have been chosen. One such case is clustering under stability of Awasthi et al. [Awasthi et al., 2010] where the number of such bad clusters is a constant. Using property (i), we significantly improve the running time of their algorithm from O(dn³) (k log{n})^{poly(1/(β), 1/(ε))} to O (dn³ (k/(ε)) ^{O(1/βε²)}). Another application is a parameterised reduction from the outlier version of k-means to the classical one where the bad clusters are the outliers. 2) Streaming algorithms: The sampling algorithm running in a single iteration (i.e., property (ii)) allows us to design a constant-pass, logspace streaming algorithm for the list-k-means problem. This can be converted to a constant-pass, logspace streaming PTAS for various constrained versions of the k-means problem. In particular, this gives a 3-pass, polylog-space streaming PTAS for the constrained binary k-means problem which in turn gives a 4-pass, polylog-space streaming PTAS for the generalised binary 𝓁₀-rank-r approximation problem. This is the first constant pass, polylog-space streaming algorithm for either of the two problems. Coreset based techniques, which is another approach for designing streaming algorithms in general, is not known to work for the constrained binary k-means problem to the best of our knowledge.