On Extended Concentration Inequalities for Fast JL Embeddings of Infinite Sets

The Johnson-Lindenstrauss (JL) lemma allows subsets of a high-dimensional space to be embedded into a lower-dimensional space while approximately preserving all pairwise Euclidean distances. This important result has inspired an extensive literature, with a significant portion dedicated to constructing structured random matrices with fast matrix-vector multiplication algorithms that generate such embeddings for finite point sets. In this paper, we briefly consider fast JL embedding matrices for {\it infinite} subsets of $\mathbb{R}^d$. Prior work in this direction such as \cite{oymak2018isometric, mendelson2023column} has focused on constructing fast JL matrices $HD \in \mathbb{R}^{k \times d}$ by multiplying structured matrices with RIP(-like) properties $H \in \mathbb{R}^{k \times d}$ against a random diagonal matrix $D \in \mathbb{R}^{d \times d}$. However, utilizing RIP(-like) matrices $H$ in this fashion necessarily has the unfortunate side effect that the resulting embedding dimension $k$ must depend on the ambient dimension $d$ no matter how simple the infinite set is that one aims to embed. Motivated by this, we explore an alternate strategy for removing this $d$-dependence from $k$ herein: Extending a concentration inequality proven by Ailon and Liberty \cite{Ailon2008fast} in the hope of later utilizing it in a chaining argument to obtain a near-optimal result for infinite sets. %, and $(ii)$ utilizing a simple secondary Gaussian embedding of an initial fast JL embedding of a given infinite set. Though this strategy ultimately fails to provide the near-optimal embedding dimension we seek, along the way we obtain a stronger-than-sub-exponential extension of the concentration inequality in \cite{Ailon2008fast} which may be of independent interest.