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Sparse SVD # Singular Value Decomposition Given an *m x n* matrix *A*, compute matrices *U, S, V* such that *A = U * S * V^T* There is no restriction on m, but we require n^2 doubles to fit in memory. Further, n should be less than m. The decomposition is computed by first computing *A^TA = V S^2 V^T*, computing svd locally on that (since n x n is small), from which we recover S and V. Then we compute U via easy matrix multiplication as *U = A * V * S^-1* Only singular vectors associated with the largest k singular values If there are k such values, then the dimensions of the return will be: * *S* is *k x k* and diagonal, holding the singular values on diagonal. * *U* is *m x k* and satisfies U^T*U = eye(k). * *V* is *n x k* and satisfies V^TV = eye(k). All input and output is expected in sparse matrix format, 0-indexed as tuples of the form ((i,j),value) all in RDDs. # Testing Tests included. They test: - Decomposition promise (A = USV^T) - For small matrices, output is compared to that of jblas - Rank 1 matrix test included - Full Rank matrix test included - Middle-rank matrix forced via k included # Example Usage import org.apache.spark.SparkContext import org.apache.spark.mllib.linalg.SVD import org.apache.spark.mllib.linalg.SparseMatrix import org.apache.spark.mllib.linalg.MatrixyEntry // Load and parse the data file val data = sc.textFile("mllib/data/als/test.data").map { line => val parts = line.split(',') MatrixEntry(parts(0).toInt, parts(1).toInt, parts(2).toDouble) } val m = 4 val n = 4 // recover top 1 singular vector val decomposed = SVD.sparseSVD(SparseMatrix(data, m, n), 1) println("singular values = " + decomposed.S.data.toArray.mkString) # Documentation Added to docs/mllib-guide.md |
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