Approximation and Streaming Algorithms for Projective Clustering via Random Projections

Let $P$ be a set of $n$ points in $\mathbb{R}^d$. In the projective clustering problem, given $k, q$ and norm $ρ\in [1,\infty]$, we have to compute a set $\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\sum_{p\in P}d(p, \mathcal{F})^ρ)^{1/ρ}$ is minimized; here $d(p, \mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\mathcal{F}$. We let $f_k^q(P,ρ)$ denote the minimal value and interpret $f_k^q(P,\infty)$ to be $\max_{r\in P}d(r, \mathcal{F})$. When $ρ=1,2$ and $\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the $k$-center clustering problems respectively. For every $0 < ε< 1$, $S\subset P$ and $ρ\ge 1$, we show that the orthogonal projection of $P$ onto a randomly chosen flat of dimension $O(((q+1)^2\log(1/ε)/ε^3) \log n)$ will $ε$-approximate $f_1^q(S,ρ)$. This result combines the concepts of geometric coresets and subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence, an orthogonal projection of $P$ to an $O(((q+1)^2 \log ((q+1)/ε)/ε^3) \log n)$ dimensional randomly chosen subspace $ε$-approximates projective clusterings for every $k$ and $ρ$ simultaneously. Note that the dimension of this subspace is independent of the number of clusters~$k$. Using this dimension reduction result, we obtain new approximation and streaming algorithms for projective clustering problems. For example, given a stream of $n$ points, we show how to compute an $ε$-approximate projective clustering for every $k$ and $ρ$ simultaneously using only $O((n+d)((q+1)^2\log ((q+1)/ε))/ε^3 \log n)$ space. Compared to standard streaming algorithms with $Ω(kd)$ space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude.

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