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 $\rho \in [1,\infty]$, we have to compute a set $\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\sum_{p\in P}d(p, \mathcal{F})^\rho)^{1/\rho}$ is minimized; here $d(p, \mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\mathcal{F}$. We let $f_k^q(P,\rho)$ denote the minimal value and interpret $f_k^q(P,\infty)$ to be $\max_{r\in P}d(r, \mathcal{F})$. When $\rho=1,2$ and $\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the $k$-center clustering problems respectively. For every $0 < \epsilon < 1$, $S\subset P$ and $\rho \ge 1$, we show that the orthogonal projection of $P$ onto a randomly chosen flat of dimension $O(((q+1)^2\log(1/\epsilon)/\epsilon^3) \log n)$ will $\epsilon$-approximate $f_1^q(S,\rho)$. 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 $O(((q+1)^2 \log ((q+1)/\epsilon)/\epsilon^3) \log n)$ dimensional randomly chosen subspace $\epsilon$-approximates projective clusterings for every $k$ and $\rho$ 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 $\epsilon$-approximate projective clustering for every $k$ and $\rho$ simultaneously using only $O((n+d)((q+1)^2\log ((q+1)/\epsilon))/\epsilon^3 \log n)$ space. Compared to standard streaming algorithms with $\Omega(kd)$ space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude.Comment: Canadian Conference on Computational Geometry (CCCG 2015

A Dimension Reduction Scheme for the Computation of Optimal Unions of Subspaces

Given a set of points \F in a high dimensional space, the problem of finding a union of subspaces \cup_i V_i\subset \R^N that best explains the data \F increases dramatically with the dimension of \R^N. In this article, we study a class of transformations that map the problem into another one in lower dimension. We use the best model in the low dimensional space to approximate the best solution in the original high dimensional space. We then estimate the error produced between this solution and the optimal solution in the high dimensional space.Comment: 15 pages. Some corrections were added, in particular the title was changed. It will appear in "Sampling Theory in Signal and Image Processing