We compute the singular values of an @math sparse matrix @math in a distributed setting, without communication dependence on @math , which is useful for very large @math . In particular, we give a simple nonadaptive sampling scheme where the singular values of @math are estimated within relative error with constant probability. Our proven bounds focus on the MapReduce framework, which has become the de facto tool for handling such large matrices that cannot be stored or even streamed through a single machine. On the way, we give a general method to compute @math . We preserve singular values of @math with @math relative error with shuffle size @math and reduce-key complexity @math . We further show that if only specific entries of @math are required and @math has nonnegative entries, then we can reduce the shuffle size to @math and reduce-key complexity to @math , where @math is the minimum cosine similarity for the entries being estimated. All of our bounds are independent of @math , the larger dimension. We provide open-source implementations in Spark and Scalding, along with experiments in an industrial setting.
Recently several results appeared that show significant reduction in time for matrix multiplication, singular value decomposition as well as linear ( 2) regression, all based on data dependent random sampling. Our key idea is that low dimensional embeddings can be used to eliminate data dependence and provide more versatile, linear time pass efficient matrix computation. Our main contribution is summarized as follows. --Independent of the recent results of Har-Peled and of Deshpande and Vempala, one of the first -- and to the best of our knowledge the most efficient -- relative error (1 + ) A - A_k _F approximation algorithms for the singular value decomposition of an m ? n matrix A with M non-zero entries that requires 2 passes over the data and runs in time O ( ( M( k + k k) + (n + m)( k + k k)^2 ) 1 ) --The first o(nd^ 2 ) time (1 + ) relative error approximation algorithm for n ? d linear ( ) regression. --A matrix multiplication and norm approximation algorithm that easily applies to implicitly given matrices and can be used as a black box probability boosting tool. 1 Introduction 2 Map 3 The Data Stream Phenomenon 4 Data Streaming: Formal Aspects 5 Foundations: Basic Mathematical Ideas 6 Foundations: Basic Algorithmic Techniques 7 Foundations: Summary 8 Streaming Systems 9 New Directions 10 Historic Notes 11 Concluding Remarks Acknowledgements References.