Dimensionality Reduction for k-Means Clustering and Low Rank Approximation

Abstract

We show how to approximate a data matrix A with a much smaller sketch ~A that can be used to solve a general class of constrained k-rank approximation problems to within (1+ε) error. Importantly, this class includes k-means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just O(k) dimensions, we generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For k-means dimensionality reduction, we provide (1+ε) relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only 'cover' a good subspace for A}, but can be used directly to compute this subspace. Finally, for k-means clustering, we show how to achieve a (9+ε) approximation by Johnson-Lindenstrauss projecting data to just O(log k/ε2) dimensions. This is the first result that leverages the specific structure of k-means to achieve dimension independent of input size and sublinear in k.