本文原作者：尹迪，经授权后发布。
假设向量v是方阵A的特征向量，可以表示成下面的形式：
这里lambda表示特征向量v所对应的特征值。并且一个矩阵的一组特征向量是一组正交向量。特征值分解是将一个矩阵分解为下面的形式：
其中Q是这个矩阵A的特征向量组成的矩阵。sigma是一个对角矩阵，每个对角线上的元素就是一个特征值。
特征值分解是一个提取矩阵特征很不错的方法，但是它只适合于方阵，对于非方阵，它不适合。这就需要用到奇异值分解。
1 源码分析
MLlib使用ARPACK来求解特征值分解。它的实现代码如下
def symmetricEigs(
mul: BDV[Double] => BDV[Double],
n: Int,
k: Int,
tol: Double,
maxIterations: Int): (BDV[Double], BDM[Double]) = {
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)
}
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)
} 我们可以查看ARPACK的注释详细了解dsaupd和dseupd方法的作用。
