Faster Coreset Construction for Projective Clustering via Low-Rank Approximation

TL;DR

This paper introduces a faster randomized coreset construction method for projective clustering that leverages low-rank approximation, reducing computational complexity and enabling efficient streaming implementation for large datasets.

Contribution

A novel randomized coreset construction method using low-rank approximation that improves speed and efficiency over previous deterministic approaches for projective clustering.

Findings

Achieves faster coreset construction for small k and j.

Maintains similar approximation quality to existing methods.

Efficient in streaming and sparse data scenarios.

Abstract

In this work, we present a randomized coreset construction for projective clustering, which involves computing a set of $k$ closest $j$-dimensional linear (affine) subspaces of a given set of $n$ vectors in $d$ dimensions. Let $A \in \mathbb{R}^{n\times d}$ be an input matrix. An earlier deterministic coreset construction of Feldman \textit{et. al.} relied on computing the SVD of $A$. The best known algorithms for SVD require $\min\{nd^2, n^2d\}$ time, which may not be feasible for large values of $n$ and $d$. We present a coreset construction by projecting the rows of matrix $A$ on some orthonormal vectors that closely approximate the right singular vectors of $A$. As a consequence, when the values of $k$ and $j$ are small, we are able to achieve a faster algorithm, as compared to the algorithm of Feldman \textit{et. al.}, while maintaining almost the same approximation. We also benefit in terms of space as well as exploit the sparsity of the input dataset. Another advantage of our approach is that it can be constructed in a streaming setting quite efficiently.