A variant of the Johnson-Lindenstrauss lemma for circulant matrices

Abstract

We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in \cite{HV}. We reduce the bound on $k$ from $k=O({\epsilon }^{-2}{\log }^{3}n)$ proven there to $k=O({\epsilon }^{-2}{\log }^{2}n)$. Our technique differs essentially from the one used in \cite{HV}. We employ the discrete Fourier transform and singular value decomposition to deal with the dependency caused by the circulant structure.