# Dimensionality Reductions That Preserve Volumes and Distance to Affine Spaces, and Their Algorithmic Applications

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## Abstract

Let *X* be a subset of *n* points of the Euclidean space, and let 0 < ɛ < 1. A classical result of Johnson and Lindenstrauss [JL84] states that there is a projection of *X* onto a subspace of dimension O(ɛ^{-2} log *n*), with distortion ≤ 1 + ɛ. Here we show a natural extension of the above result, to a stronger preservation of the geometry of finite spaces. By a *k*-fold increase of the number of dimensions used compared to [JL84], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show it is possible to embed a subset of size *n* of the Euclidean space into a *O*(ɛ^{-2} *k*log*n*)-dimensional Euclidean space, so that no set of size *s* ≤ *k* changes its volume by more than (1+ɛ)^{s-1}. Moreover, distances of points from affine hulls of sets of at most *k*-1 points in the space do not change by more than a factor of 1 + ɛ. A consequence of the above with *k* = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL84]. Our method can be applied to many problems with high-dimensional nature such as *Projective Clustering* and *Approximated Nearest Affine Neighbor Search*. In particular, it shows a first poly-logarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [Fei00], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.

## Keywords

Euclidean Space Dimensionality Reduction Query Point Random Projection Dimensional Euclidean Space## Preview

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