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

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 poly(k/ε). A strong coreset (1+ε)-approximates the cost function for all possible sets of centers simultaneously. We also give efficient nnz(A) + (n+d) poly(k/ε) + exp(poly(k/ε)) 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 [FSS13] for squared Euclidean distances to sums of P-th powers of Euclidean distances for constant p≥1.