High-Dimensional Shape Fitting in Linear Time

Let $P$ be a set of $n$ points in $\Re^d$. The {\em radius} of a $k$-dimensional flat ${\cal F}$ with respect to $P$, which we denote by ${\cal RD}({\cal F},P)$, is defined to be $\max_{p \in P} \mathop{\rm dist}({\cal F},p)$, where $\mathop{\rm dist}({\cal F},p)$ denotes the Euclidean distance between $p$ and its projection onto ${\cal F}$. The $k$-flat radius of $P$, which we denote by ${R^{\rm opt}_k}(P)$, is the minimum, over all $k$-dimensional flats ${\cal F}$, of ${\cal RD}({\cal F},P)$. We consider the problem of computing ${R^{\rm opt}_k}(P)$ 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 $0 < \eps \leq 1$, returns a $k$-flat ${\cal F}$ such that ${\cal RD}({\cal F},P) \leq (1 + \eps) {R^{\rm opt}_k}(P)$. The algorithm runs in $O(nd C_{\eps,k})$ time, where $C_{\eps,k}$ is a constant that depends only on $\eps$ 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 $d n^{O(k/\eps^c)}$, where $c$ is an appropriate constant.