High-dimensional shape fitting in linear time

Let P be a set of n points in ℝ^d. The radius of a k-dimensional flat F with respect to P, denoted by RD(F, P), is defined to be max_p∈P dist(F, p), where dist(F, p) denotes the Euclidean distance between p and its projection onto F. The k-flat radius of P, which we denote by R_k^opt(P), is the minimum, over all k-dimensional flats F, of RD(F, P). We consider the problem of computing R_k^opt(P) for a given set of points P. We are interested in the high-dimensional case where d is a part of the input and not a constant. This problem is ℕℙ-hard even for k = 1. We present an algorithm that, given P and a parameter 0 < ε ≤ 1, returns a k-flat F such that RD(F, P) ≤ (1 + ε)R_k^opt(P). The algorithm runs in O(ndC_ε,k) time, where C_ε,k is a constant that depends only on ε and k. Thus the algorithm runs in time linear in the size of the point set and is a substantial improvement over previous known algorithms, whose running time is of the order of dn^O(k/εc) where c is an appropriate constant.