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

All tests passed. I will run it against some large matrices.

Author: Li Pu <[email protected]>
Author: Xiangrui Meng <[email protected]>
Author: Li Pu <[email protected]>

Closes #964 from vrilleup/master and squashes the following commits:

7312ec1 [Li Pu] very minor comment fix
4c618e9 [Li Pu] Merge pull request #1 from mengxr/vrilleup-master
a461082 [Xiangrui Meng] make superscript show up correctly in doc
861ec48 [Xiangrui Meng] simplify axpy
62969fa [Xiangrui Meng] use BDV directly in symmetricEigs change the computation mode to local-svd, local-eigs, and dist-eigs update tests and docs
c273771 [Li Pu] automatically determine SVD compute mode and parameters
7148426 [Li Pu] improve RowMatrix multiply
5543cce [Li Pu] improve svd api
819824b [Li Pu] add flag for dense svd or sparse svd
eb15100 [Li Pu] fix binary compatibility
4c7aec3 [Li Pu] improve comments
e7850ed [Li Pu] use aggregate and axpy
827411b [Li Pu] fix EOF new line
9c80515 [Li Pu] use non-sparse implementation when k = n
fe983b0 [Li Pu] improve scala style
96d2ecb [Li Pu] improve eigenvalue sorting
e1db950 [Li Pu] SPARK-1782: svd for sparse matrix using ARPACK

Author: Li Pu

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

@@ -0,0 +1,157 @@
+/*
+ * 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 breeze.linalg.{DenseMatrix => BDM, DenseVector => BDV}
+import com.github.fommil.netlib.ARPACK
+import org.netlib.util.{intW, doubleW}
+
+import org.apache.spark.annotation.Experimental
+
+/**
+ * :: Experimental ::
+ * Compute eigen-decomposition.
+ */
+@Experimental
+private[mllib] 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 DenseVector.
+   * @param n dimension of the square matrix (maximum Int.MaxValue).
+   * @param k number of leading eigenvalues required, 0 < k < n.
+   * @param tol tolerance of the eigs computation.
+   * @param maxIterations the maximum number of Arnoldi update iterations.
+   * @return a dense vector of eigenvalues in descending order and a dense matrix of eigenvectors
+   *         (columns of the matrix).
+   * @note The number of computed eigenvalues might be smaller than k when some Ritz values do not
+   *       satisfy the convergence criterion specified by tol (see ARPACK Users Guide, Chapter 4.6
+   *       for more details). The maximum number of Arnoldi update iterations is set to 300 in this
+   *       function.
+   */
+  private[mllib] def symmetricEigs(
+      mul: BDV[Double] => BDV[Double],
+      n: Int,
+      k: Int,
+      tol: Double,
+      maxIterations: Int): (BDV[Double], BDM[Double]) = {
+    // TODO: remove this function and use eigs in breeze when switching breeze version
+    require(n > k, s"Number of required eigenvalues $k must be smaller than matrix dimension $n")
+
+    val arpack = ARPACK.getInstance()
+
+    // tolerance used in stopping criterion
+    val tolW = new doubleW(tol)
+    // number of desired eigenvalues, 0 < nev < n
+    val nev = new intW(k)
+    // nev Lanczos vectors are generated in the first iteration
+    // ncv-nev Lanczos vectors are generated in each subsequent iteration
+    // ncv must be smaller than n
+    val ncv = math.min(2 * k, n)
+
+    // "I" for standard eigenvalue problem, "G" for generalized eigenvalue problem
+    val bmat = "I"
+    // "LM" : compute the NEV largest (in magnitude) eigenvalues
+    val which = "LM"
+
+    var iparam = new Array[Int](11)
+    // use exact shift in each iteration
+    iparam(0) = 1
+    // maximum number of Arnoldi update iterations, or the actual number of iterations on output
+    iparam(2) = maxIterations
+    // Mode 1: A*x = lambda*x, A symmetric
+    iparam(6) = 1
+
+    var ido = new intW(0)
+    var info = new intW(0)
+    var resid = new Array[Double](n)
+    var v = new Array[Double](n * ncv)
+    var workd = new Array[Double](n * 3)
+    var workl = new Array[Double](ncv * (ncv + 8))
+    var ipntr = new Array[Int](11)
+
+    // call ARPACK's reverse communication, first iteration with ido = 0
+    arpack.dsaupd(ido, bmat, n, which, nev.`val`, tolW, resid, ncv, v, n, iparam, ipntr, workd,
+      workl, workl.length, info)
+
+    val w = BDV(workd)
+
+    // ido = 99 : done flag in reverse communication
+    while (ido.`val` != 99) {
+      if (ido.`val` != -1 && ido.`val` != 1) {
+        throw new IllegalStateException("ARPACK returns ido = " + ido.`val` +
+            " This flag is not compatible with Mode 1: A*x = lambda*x, A symmetric.")
+      }
+      // multiply working vector with the matrix
+      val inputOffset = ipntr(0) - 1
+      val outputOffset = ipntr(1) - 1
+      val x = w.slice(inputOffset, inputOffset + n)
+      val y = w.slice(outputOffset, outputOffset + n)
+      y := mul(x)
+      // call ARPACK's reverse communication
+      arpack.dsaupd(ido, bmat, n, which, nev.`val`, tolW, resid, ncv, v, n, iparam, ipntr,
+        workd, workl, workl.length, info)
+    }
+
+    if (info.`val` != 0) {
+      info.`val` match {
+        case 1 => throw new IllegalStateException("ARPACK returns non-zero info = " + info.`val` +
+            " Maximum number of iterations taken. (Refer ARPACK user guide for details)")
+        case 2 => throw new IllegalStateException("ARPACK returns non-zero info = " + info.`val` +
+            " No shifts could be applied. Try to increase NCV. " +
+            "(Refer ARPACK user guide for details)")
+        case _ => throw new IllegalStateException("ARPACK returns non-zero info = " + info.`val` +
+            " Please refer ARPACK user guide for error message.")
+      }
+    }
+
+    val d = new Array[Double](nev.`val`)
+    val select = new Array[Boolean](ncv)
+    // copy the Ritz vectors
+    val z = java.util.Arrays.copyOfRange(v, 0, nev.`val` * n)
+
+    // call ARPACK's post-processing for eigenvectors
+    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)
+
+    // number of computed eigenvalues, might be smaller than k
+    val computed = iparam(4)
+
+    val eigenPairs = java.util.Arrays.copyOfRange(d, 0, computed).zipWithIndex.map { r =>
+      (r._1, java.util.Arrays.copyOfRange(z, r._2 * n, r._2 * n + n))
+    }
+
+    // sort the eigen-pairs in descending order
+    val sortedEigenPairs = eigenPairs.sortBy(- _._1)
+
+    // copy eigenvectors in descending order of eigenvalues
+    val sortedU = BDM.zeros[Double](n, computed)
+    sortedEigenPairs.zipWithIndex.foreach { r =>
+      val b = r._2 * n
+      var i = 0
+      while (i < n) {
+        sortedU.data(b + i) = r._1._2(i)
+        i += 1
+      }
+    }
+
+    (BDV[Double](sortedEigenPairs.map(_._1)), sortedU)
+  }
+}