Efficient methods for grouping vectors into low-rank clusters

We present a few practical algorithms for sorting vectors into low-rank clusters. These algorithms rely on a subdivision scheme applied to the space of projections from d-dimensions to 1-dimension. This subdivision scheme can be thought of as a higher-dimensional generalization of quicksort. Given the ability to quickly sort vectors into low-rank clusters, one can efficiently search a matrix for low-rank sub-blocks of large diameter. The ability to detect large-diameter low-rank sub-blocks has many applications, ranging from data-analysis to matrix-compression.