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 $p\ge 1$.

References in corpus (1)

• Input Sparsity and Hardness for Robust Subspace Approximation

Cited by in corpus (1)

• Coresets for Near-Convex Functions