Bi-criteria linear-time approximations for generalized k-mean/median/center

We consider the problem of approximating a set P of n points in R ^d by a collection of j-dimensional flats, andextensions thereof, under the standard median / mean / centermeasures, in which we wish to minimize, respectively, the sum of thedistances from each point of P to its nearest flat, the sum of thesquares of these distances, or the maximal such distance.Such problems cannot be approximated unless P=NP but do allowbi-criteria approximations where one allows some leeway in both the numberof flats and the quality of the objective function.We give a very simple bi-criteria approximation algorithm, which producesat most α(k,j,n) = (k j log n) ^O(j) flats, which exceeds the optimalobjective value for any k j-dimensional flats by a factor of nomore than Β(j)= 2^O(j). Given this bi-criteria approximation, wecan use it to reduce the approximation factor arbitrarily, at the costof increasing the number of flats. Our algorithm hasmany advantages over previous work, in that it is muchmore widely applicable (wider set of objective functions and classes ofclusters) and much more efficient - reducing the running time bound from O(n^Poly(k,j)) to nd (jk)^O(j).Our algorithm is randomized and successful with probability 1/2(easily boosted to probabilities arbitrary close to 1).