Faster Projective Clustering Approximation of Big Data

TL;DR

This paper introduces a faster approximation method for projective clustering in big data, significantly reducing coreset size and computation time while handling outliers, with theoretical guarantees and experimental validation.

Contribution

It presents the first $O( ext{log}(m))$ approximation algorithm for $m$ lines clustering, reducing coreset size from exponential to logarithmic in $m$, and extends to outlier handling.

Findings

Achieves $O( ext{log}(m))$ approximation in $O(ndm)$ time.

Provides a coreset construction for projective clustering.

Includes experimental results and open-source implementation.

Abstract

In projective clustering we are given a set of n points in $R^d$ and wish to cluster them to a set $S$ of $k$ linear subspaces in $R^d$ according to some given distance function. An $\eps$-coreset for this problem is a weighted (scaled) subset of the input points such that for every such possible $S$ the sum of these distances is approximated up to a factor of $(1+\eps)$. We suggest to reduce the size of existing coresets by suggesting the first $O(\log(m))$ approximation for the case of $m$ lines clustering in $O(ndm)$ time, compared to the existing $\exp(m)$ solution. We then project the points on these lines and prove that for a sufficiently large $m$ we obtain a coreset for projective clustering. Our algorithm also generalize to handle outliers. Experimental results and open code are also provided.