Euclidean Embeddings that Preserve Volumes

Let P be a set of n points in Euclidean space and let 0 < ε < 1. A well-known result of Johnson and Lindenstrauss states that there is a projection of P onto a subspace of dimension O(ε−2 logn) such that distances change by a factor of 1 + ε at most. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs, but all subsets of at most k points maintain their volume approximately. More precisely, we require that sets of size s ≤ k preserve their volumes within a factor of (1 + ε)s−1. We show that this can be achieved using O(max{k/ε,ε−2 logn}) dimensions. This in particular means that for k = O(logn/ε) we require no more dimensions (asymptotically) than the special case k = 2, handled by Johnson and Lindenstrauss. Our work improves on a result of Magen that required as many as O(kε−2 logn) dimensions and is tight up to a factor of O(1/ε). Another outcome of our work is an alternative and simplified proof of the result of Magen showing that all distances between points and affine subspaces spanned by a small number of points are approximately preserved when projecting onto O(kε−2 logn) dimensions. Instead of fixing the distortion and finding the required dimension, we also consider