Large-scale Submodular Greedy Exemplar Selection with Structured Similarity Matrices

Exemplar clustering attempts to find a subset of data-points that summarizes the entire data-set in the sense of minimizing the sum of distances from each point to its closest exemplar. It has many important applications in machine learning including document and video summarization, data compression, scalability of kernel methods and Gaussian processes, active learning and feature selection. A key challenge in the adoption of exemplar clustering to large-scale applications has been the availability of accurate and scalable algorithms. We propose an approach that combines structured similarity matrix representations with submodular greedy maximization that can dramatically increase the scalability of exemplar clustering and still enjoys good approximation guarantees. Exploiting structured similarity matrices within the context of submodular greedy algorithms is by no means trivial, as naive approaches still require computing all the entries of the matrix. We propose a randomized approach based on sampling sign-patterns of columns of the similarity matrix and establish accuracy guarantees. We demonstrate significant computational speed-ups while still achieving highly accurate solutions, and solve problems with up-to millions of data-points in around a minute or less on a single commodity computer.