Block CUR : Decomposing Large Distributed Matrices
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A common problem in large-scale data analysis is to approximate a matrix using a combination of specifically sampled rows and columns, known as CUR decomposition. Unfortunately, in many real-world environments, the ability to sample specific individual rows or columns of the matrix is limited by either system constraints or cost. In this paper, we consider matrix approximation by sampling predefined blocks of columns (or rows) from the matrix. This regime is commonly found when data is distributed across multiple nodes in a compute cluster, where such blocks correspond to columns (or rows) of the matrix stored on the same node, which can be retrieved with much less overhead than retrieving individual columns stored across different nodes. We propose a novel algorithm for sampling useful column blocks and provide guarantees for the quality of the approximation. We demonstrate the practical utility of this algorithm for computing the block CUR decomposition of large matrices in a distributed setting using Apache Spark. Using our proposed block CUR algorithms, we can achieve a significant speed-up compared to a regular CUR decomposition with the same quality of approximation.
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