Near Optimal Dimensionality Reductions That Preserve Volumes

Abstract

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 $\mathcal{O}(\epsilon^{-2} \log n)$ such that distances change by at most a factor of 1 + ε. 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 $\mathcal{O}(\max\{\frac{k}{\epsilon},\epsilon^{-2}\log n\})$ dimensions. This in particular means that for $k = \mathcal{O}(\log n/\epsilon)$ 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 $\mathcal{O}(k\epsilon^{-2}\log n)$ dimensions) and is tight up to a factor of $\mathcal{O}(1/\epsilon)$. Another outcome of our work is an alternative and greatly 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 $\mathcal{O}(k\epsilon^{-2}\log n)$ dimensions.