[Spark-5406][MLlib] LocalLAPACK mode in RowMatrix.computeSVD should h…

…ave much smaller upper bound

JIRA link: https://issues.apache.org/jira/browse/SPARK-5406

The code in breeze svd  imposes the upper bound for LocalLAPACK in RowMatrix.computeSVD
code from breeze svd (https://github.com/scalanlp/breeze/blob/master/math/src/main/scala/breeze/linalg/functions/svd.scala)
     val workSize = ( 3
        * scala.math.min(m, n)
        * scala.math.min(m, n)
        + scala.math.max(scala.math.max(m, n), 4 * scala.math.min(m, n)
          * scala.math.min(m, n) + 4 * scala.math.min(m, n))
      )
      val work = new Array[Double](workSize)

As a result, 7 * n * n + 4 * n < Int.MaxValue at least (depends on JVM)

In some worse cases, like n = 25000, work size will become positive again (80032704) and bring wired behavior.

The PR is only the beginning, to support Genbase ( an important biological benchmark that would help promote Spark to genetic applications, http://www.paradigm4.com/wp-content/uploads/2014/06/Genomics-Benchmark-Technical-Report.pdf),
which needs to compute svd for matrix up to 60K * 70K. I found many potential issues and would like to know if there's any plan undergoing that would expand the range of matrix computation based on Spark.
Thanks.

Author: Yuhao Yang <[email protected]>

Closes apache#4200 from hhbyyh/rowMatrix and squashes the following commits:

f7864d0 [Yuhao Yang] update auto logic for rowMatrix svd
23860e4 [Yuhao Yang] fix comment style
e48a6e4 [Yuhao Yang] make latent svd computation constraint clear

mllib/src/main/scala/org/apache/spark/mllib/linalg/distributed/RowMatrix.scala

@@ -219,8 +219,12 @@ class RowMatrix(
 
     val computeMode = mode match {
       case "auto" =>
+        if(k > 5000) {
+          logWarning(s"computing svd with k=$k and n=$n, please check necessity")
+        }
+
         // TODO: The conditions below are not fully tested.
-        if (n < 100 || k > n / 2) {
+        if (n < 100 || (k > n / 2 && n <= 15000)) {
           // If n is small or k is large compared with n, we better compute the Gramian matrix first
           // and then compute its eigenvalues locally, instead of making multiple passes.
           if (k < n / 3) {
@@ -245,6 +249,8 @@ class RowMatrix(
         val G = computeGramianMatrix().toBreeze.asInstanceOf[BDM[Double]]
         EigenValueDecomposition.symmetricEigs(v => G * v, n, k, tol, maxIter)
       case SVDMode.LocalLAPACK =>
+        // breeze (v0.10) svd latent constraint, 7 * n * n + 4 * n < Int.MaxValue
+        require(n < 17515, s"$n exceeds the breeze svd capability")
         val G = computeGramianMatrix().toBreeze.asInstanceOf[BDM[Double]]
         val brzSvd.SVD(uFull: BDM[Double], sigmaSquaresFull: BDV[Double], _) = brzSvd(G)
         (sigmaSquaresFull, uFull)