SPARK-1782: svd for sparse matrix using ARPACK

copy ARPACK dsaupd/dseupd code from latest breeze
change RowMatrix to use sparse SVD
change tests for sparse SVD

Author: Li Pu

mllib/src/main/scala/org/apache/spark/mllib/linalg/EigenValueDecomposition.scala

@@ -0,0 +1,120 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.spark.mllib.linalg
+
+import org.apache.spark.annotation.Experimental
+import breeze.linalg.{DenseMatrix => BDM, DenseVector => BDV}
+import org.netlib.util.{intW, doubleW}
+import com.github.fommil.netlib.ARPACK
+
+/**
+ * :: Experimental ::
+ * Represents eigenvalue decomposition factors.
+ */
+@Experimental
+case class EigenValueDecomposition[VType](s: Vector, V: VType)
+
+object EigenValueDecomposition {
+  /**
+   * Compute the leading k eigenvalues and eigenvectors on a symmetric square matrix using ARPACK.
+   * The caller needs to ensure that the input matrix is real symmetric. This function requires
+   * memory for `n*(4*k+4)` doubles.
+   *
+   * @param mul a function that multiplies the symmetric matrix with a Vector.
+   * @param n dimension of the square matrix (maximum Int.MaxValue).
+   * @param k number of leading eigenvalues required.
+   * @param tol tolerance of the eigs computation.
+   * @return a dense vector of eigenvalues in descending order and a dense matrix of eigenvectors
+   *         (columns of the matrix). The number of computed eigenvalues might be smaller than k.
+   */
+  private[mllib] def symmetricEigs(mul: Vector => Vector, n: Int, k: Int, tol: Double)
+    : (BDV[Double], BDM[Double]) = {
+    require(n > k, s"Number of required eigenvalues $k must be smaller than matrix dimension $n")
+
+    val arpack  = ARPACK.getInstance()
+
+    val tolW = new doubleW(tol)
+    val nev = new intW(k)
+    val ncv = scala.math.min(2*k,n)
+
+    val bmat = "I"
+    val which = "LM"
+
+    var iparam = new Array[Int](11)
+    iparam(0) = 1
+    iparam(2) = 300
+    iparam(6) = 1
+
+    var ido = new intW(0)
+    var info = new intW(0)
+    var resid:Array[Double] = new Array[Double](n)
+    var v = new Array[Double](n*ncv)
+    var workd = new Array[Double](3*n)
+    var workl = new Array[Double](ncv*(ncv+8))
+    var ipntr = new Array[Int](11)
+
+    // first call to ARPACK
+    arpack.dsaupd(ido, bmat, n, which, nev.`val`, tolW, resid, ncv, v, n, iparam, ipntr, workd,
+      workl, workl.length, info)
+
+    val w = BDV(workd)
+
+    while(ido.`val` !=99) {
+      if (ido.`val` != -1 && ido.`val` != 1)
+        throw new IllegalStateException("ARPACK returns ido = " + ido.`val`)
+      // multiply working vector with the matrix
+      val inputOffset = ipntr(0) - 1
+      val outputOffset = ipntr(1) - 1
+      val x = w(inputOffset until inputOffset + n)
+      val y = w(outputOffset until outputOffset + n)
+      y := BDV(mul(Vectors.fromBreeze(x)).toArray)
+      // call ARPACK
+      arpack.dsaupd(ido, bmat, n, which, nev.`val`, tolW, resid, ncv, v, n, iparam, ipntr,
+        workd, workl, workl.length, info)
+    }
+
+    if (info.`val` != 0)
+      throw new IllegalStateException("ARPACK returns non-zero info = " + info.`val`)
+
+    val d = new Array[Double](nev.`val`)
+    val select = new Array[Boolean](ncv)
+    val z = java.util.Arrays.copyOfRange(v, 0, nev.`val` * n)
+
+    arpack.dseupd(true, "A", select, d, z, n, 0.0, bmat, n, which, nev, tol, resid, ncv, v, n,
+      iparam, ipntr, workd, workl, workl.length, info)
+
+    val computed = iparam(4)
+
+    val s = BDV(d)(0 until computed)
+    val U = new BDM(n, computed, z)
+
+    val sortedEigenValuesWithIndex = s.toArray.zipWithIndex.sortBy(-1 * _._1).zipWithIndex
+
+    val sorteds = BDV(sortedEigenValuesWithIndex.map(_._1._1))
+    val sortedU = BDM.zeros[Double](n, computed)
+
+    // copy eigenvectors in descending order of eigenvalues
+    sortedEigenValuesWithIndex.map{
+      r => {
+        sortedU(::, r._2) := U(::, r._1._2)
+      }
+    }
+
+    (sorteds, sortedU)
+  }
+}

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

@@ -200,6 +200,19 @@ class RowMatrix(
     nRows
   }
 
+  /**
+   * Multiply the Gramian matrix `A^T A` by a Vector on the right.
+   *
+   * @param v a local vector whose length must match the number of columns of this matrix
+   * @return a local DenseVector representing the product
+   */
+  private[mllib] def multiplyGramianMatrix(v: Vector): Vector = {
+    val bv = rows.map{
+      row => row.toBreeze * row.toBreeze.dot(v.toBreeze)
+    }.reduce( (x: BV[Double], y: BV[Double]) => x + y )
+    Vectors.fromBreeze(bv)
+  }
+
   /**
    * Computes the Gramian matrix `A^T A`.
    */
@@ -221,15 +234,17 @@ class RowMatrix(
 
   /**
    * Computes the singular value decomposition of this matrix.
-   * Denote this matrix by A (m x n), this will compute matrices U, S, V such that A = U * S * V'.
+   * Denote this matrix by A (m x n), this will compute matrices U, S, V such that A ~= U * S * V',
+   * where S contains the leading singular values, U and V contain the corresponding singular
+   * vectors.
    *
-   * 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'A = V S^2 V',
-   * 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).
-   * Note that this approach requires `O(n^3)` time on the master node.
+   * There is no restriction on m, but we require `n*(6*k+4)` doubles to fit in memory on the master
+   * node. Further, n should be less than m.
+   *
+   * The decomposition is computed by providing a function that multiples a vector with A'A to
+   * ARPACK, and iteratively invoking ARPACK-dsaupd on master node, from which we recover S and V.
+   * Then we compute U via easy matrix multiplication as U =  A * (V * S-1).
+   * Note that this approach requires `O(nnz(A))` time.
    *
    * At most k largest non-zero singular values and associated vectors are returned.
    * If there are k such values, then the dimensions of the return will be:
@@ -243,20 +258,19 @@ class RowMatrix(
    * @param computeU whether to compute U
    * @param rCond the reciprocal condition number. All singular values smaller than rCond * sigma(0)
    *              are treated as zero, where sigma(0) is the largest singular value.
+   * @param tol the tolerance of the svd computation.
    * @return SingularValueDecomposition(U, s, V)
    */
   def computeSVD(
       k: Int,
       computeU: Boolean = false,
-      rCond: Double = 1e-9): SingularValueDecomposition[RowMatrix, Matrix] = {
+      rCond: Double = 1e-9,
+      tol: Double = 1e-6): SingularValueDecomposition[RowMatrix, Matrix] = {
     val n = numCols().toInt
     require(k > 0 && k <= n, s"Request up to n singular values k=$k n=$n.")
 
-    val G = computeGramianMatrix()
-
-    // TODO: Use sparse SVD instead.
-    val (u: BDM[Double], sigmaSquares: BDV[Double], v: BDM[Double]) =
-      brzSvd(G.toBreeze.asInstanceOf[BDM[Double]])
+    val (sigmaSquares: BDV[Double], u: BDM[Double]) =
+      EigenValueDecomposition.symmetricEigs(multiplyGramianMatrix, n, k, tol)
     val sigmas: BDV[Double] = brzSqrt(sigmaSquares)
 
     // Determine effective rank.

mllib/src/test/scala/org/apache/spark/mllib/linalg/distributed/RowMatrixSuite.scala

@@ -99,7 +99,7 @@ class RowMatrixSuite extends FunSuite with LocalSparkContext {
       val localMat = mat.toBreeze()
       val (localU, localSigma, localVt) = brzSvd(localMat)
       val localV: BDM[Double] = localVt.t.toDenseMatrix
-      for (k <- 1 to n) {
+      for (k <- 1 to (n - 1)) {
         val svd = mat.computeSVD(k, computeU = true)
         val U = svd.U
         val s = svd.s
@@ -113,19 +113,19 @@ class RowMatrixSuite extends FunSuite with LocalSparkContext {
         assertColumnEqualUpToSign(V.toBreeze.asInstanceOf[BDM[Double]], localV, k)
         assert(closeToZero(s.toBreeze.asInstanceOf[BDV[Double]] - localSigma(0 until k)))
       }
-      val svdWithoutU = mat.computeSVD(n)
+      val svdWithoutU = mat.computeSVD(n - 1)
       assert(svdWithoutU.U === null)
     }
   }
 
   test("svd of a low-rank matrix") {
-    val rows = sc.parallelize(Array.fill(4)(Vectors.dense(1.0, 1.0)), 2)
-    val mat = new RowMatrix(rows, 4, 2)
+    val rows = sc.parallelize(Array.fill(4)(Vectors.dense(1.0, 1.0, 1.0)), 2)
+    val mat = new RowMatrix(rows, 4, 3)
     val svd = mat.computeSVD(2, computeU = true)
     assert(svd.s.size === 1, "should not return zero singular values")
     assert(svd.U.numRows() === 4)
     assert(svd.U.numCols() === 1)
-    assert(svd.V.numRows === 2)
+    assert(svd.V.numRows === 3)
     assert(svd.V.numCols === 1)
   }