The Johnson-Lindenstrauss Lemma for Clustering and Subspace Approximation: From Coresets to Dimension Reduction

Abstract: We study the effect of Johnson-Lindenstrauss transforms in various projective clustering problems, generalizing recent results which only applied to center-based clustering [MMR19]. We ask the general question: for a Euclidean optimization problem and an accuracy parameter $\epsilon \in (0, 1)$, what is the smallest target dimension $t \in \mathbb{N}$ such that a Johnson-Lindenstrauss transform $$\Pi \colon \mathbb{R}^d \to \mathbb{R}<sup>t$$ preserves the cost of the optimal solution up to a $(1+\epsilon)$-factor. We give a new technique which uses coreset constructions to analyze the effect of the Johnson-Lindenstrauss transform. Our technique, in addition applying to center-based clustering, improves on (or is the first to address) other Euclidean optimization problems, including: $\bullet$ For $(k,z)$-subspace approximation: we show that $t = \tilde{O}(zk² / \epsilon<sup>3)$ suffices, whereas the prior best bound, of $O(k/\epsilon<sup>2)$, only applied to the case $z = 2$ [CEMMP15]. $\bullet$ For $(k,z)$-flat approximation: we show $t = \tilde{O}(zk<sup>2/\epsilon<sup>3)$ suffices, completely removing the dependence on $n$ from the prior bound $\tilde{O}(zk² \log n/\epsilon<sup>3)$ of [KR15]. $\bullet$ For $(k,z)$-line approximation: we show $$t = O((k \log \log n + z + \log(1/\epsilon)) / \epsilon<sup>3)$$ suffices, and ours is the first to give any dimension reduction result.