Singular value decomposition

Singular value decomposition of a matrix M: m x n (real or complex) is a factorization with the form UΣV*, where U is an m x R matrix. Σ is an R x R rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an n x r unitary matrix. r is equal to the rank of the matrix M.

The diagonal entries Σii of Sigma are known as the singular values of M. The columns of U and the columns of V are called the left-singular vectors and right-singular vectors of M respectively.

The following is an example of an SVD in Apache Spark:

package linalg.svd 

import org.apache.spark.{SparkConf, SparkContext} 
import org.apache.spark.mllib.linalg.distributed.RowMatrix 
import org.apache.spark.mllib.linalg.{Matrix,       
SingularValueDecomposition, Vector, Vectors} 
object SparkSVDExampleOne { 


  def main(args: Array[String]) { 
    val denseData = Seq( 
      Vectors.dense(0.0, 1.0, 2.0, 1.0, 5.0, 3.3, 2.1), 
      Vectors.dense(3.0, 4.0, 5.0, 3.1, 4.5, 5.1, 3.3), 
      Vectors.dense(6.0, 7.0, 8.0, 2.1, 6.0, 6.7, 6.8), 
      Vectors.dense(9.0, 0.0, 1.0, 3.4, 4.3, 1.0, 1.0) 
    ) 
    val spConfig = (new 
      SparkConf).setMaster("local").setAppName("SparkSVDDemo") 
    val sc = new SparkContext(spConfig) 
    val mat: RowMatrix = new RowMatrix(sc.parallelize(denseData, 2)) 


     // Compute the top 20 singular values and corresponding    
       singular vectors. 
    val svd: SingularValueDecomposition[RowMatrix, Matrix] = 
    mat.computeSVD(7, computeU = true) 
    val U: RowMatrix = svd.U // The U factor is a RowMatrix. 
    val s: Vector = svd.s // The singular values are stored in a 
      local dense  vector. 
    val V: Matrix = svd.V // The V factor is a local dense matrix. 
    println("U:" + U) 
    println("s:" + s) 
    println("V:" + V) 
    sc.stop() 
  } 
}