A variant of the Johnson-Lindenstrauss lemma for circulant matrices

TL;DR

This paper improves the bounds for the Johnson-Lindenstrauss lemma when using circulant matrices by reducing the required embedding dimension from a cubic to a quadratic logarithmic factor, employing Fourier and SVD techniques.

Contribution

It introduces a novel approach using Fourier transform and SVD to tighten bounds for circulant matrix embeddings in Johnson-Lindenstrauss lemma.

Findings

Reduced the bound on embedding dimension from O(ε^{-2} log^3 n) to O(ε^{-2} log^2 n)

Developed a new technique employing Fourier transform and SVD for circulant matrices

Enhanced understanding of structured random matrices in dimensionality reduction

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 $k=O(\epsilon^{-2}\log^3n)$ proven there to $k=O(\epsilon^{-2}\log^2n)$. 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.