Near Optimal Dimensionality Reductions That Preserve Volumes

• Avner Magen
• Anastasios Zouzias
Conference paper

DOI: 10.1007/978-3-540-85363-3_41

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5171)
Cite this paper as:
Magen A., Zouzias A. (2008) Near Optimal Dimensionality Reductions That Preserve Volumes. In: Goel A., Jansen K., Rolim J.D.P., Rubinfeld R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg

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.