Strong Coresets for k-Median and Subspace Approximation: Goodbye Dimension

Abstract

We obtain the first strong coresets for the $k$-median and subspace approximation problems with sum of distances objective function, on $n$ points in $d$ dimensions, with a number of weighted points that is independent of both $n$ and $d$; namely, our coresets have size $\text{poly}(k/\epsilon)$. A strong coreset $(1+\epsilon)$-approximates the cost function for all possible sets of centers simultaneously. We also give efficient $\text{nnz}(A) + (n+d)\text{poly}(k/\epsilon) + \exp(\text{poly}(k/\epsilon))$ time algorithms for computing these coresets. We obtain the result by introducing a new dimensionality reduction technique for coresets that significantly generalizes an earlier result of Feldman, Sohler and Schmidt \cite{FSS13} for squared Euclidean distances to sums of $p$-th powers of Euclidean distances for constant $p\ge1$.