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

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 ε ∈ (0, 1), what is the smallest target dimension t ∈ mathbbN such that a Johnson-Lindenstrauss transform Pi colon ℝd → ℝt preserves the cost of the optimal solution up to a (1+ε)-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 = tildeO(zk2 / ε3) suffices, whereas the prior best bound, of O(k/ε2), only applied to the case z = 2 [CEMMP15]. bullet For (k,z)-flat approximation: we show t = tildeO(zk2/ε3) suffices, completely removing the dependence on n from the prior bound tildeO(zk2 log n/ε3) of [KR15]. bullet For (k,z)-line approximation: we show t = O((k log log n + z + log(1/ε)) / ε3) suffices, and ours is the first to give any dimension reduction result.