Dimensionality reductions in \(\ell_{2}\) that preserve volumes and distance to affine spaces (Q2385151)

From the author's abstract: Let \(X\) be a subset of \(n\) points of Euclidean space, and let \(0 < \varepsilon < 1\). A classical result of \textit{W. B. Johnson} and \textit{J. Lindenstrauss} [Contemp. Math. 26, 189--206 (1984; Zbl 0539.46017)] states that there is a projection of \(X\) onto a subspace of dimension \(O(\varepsilon^{-2}\log n)\) with distortion \(\leq 1+ \varepsilon\). We show a natural extension of the above result to a stronger preservation of the geometry of finite spaces. Specifically, we show how to embed a subset of size \(n\) of Euclidean space into a \(O(\varepsilon^{-2}k \log n)\)-dimensional Euclidean space, so that no set of size \(s \leq k\) changes its volume by more than \((1+\varepsilon)^{s-1}\). 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 \(1+\varepsilon\).