Github user jkbradley commented on a diff in the pull request:
https://github.com/apache/spark/pull/3871#discussion_r22423970
--- Diff:
mllib/src/main/scala/org/apache/spark/mllib/stat/impl/MultivariateGaussian.scala
---
@@ -17,23 +17,62 @@
package org.apache.spark.mllib.stat.impl
-import breeze.linalg.{DenseVector => DBV, DenseMatrix => DBM, Transpose,
det, pinv}
+import breeze.linalg.{DenseVector => DBV, DenseMatrix => DBM, max, diag,
eigSym}
-/**
- * Utility class to implement the density function for multivariate
Gaussian distribution.
- * Breeze provides this functionality, but it requires the Apache
Commons Math library,
- * so this class is here so-as to not introduce a new dependency in
Spark.
- */
+import org.apache.spark.mllib.util.MLUtils
+
+/*
+ * This class provides basic functionality for a Multivariate Gaussian
(Normal) Distribution
+ *
+ * @param mu The mean vector of the distribution
+ * @param sigma The covariance matrix of the distribution
+ */
private[mllib] class MultivariateGaussian(
val mu: DBV[Double],
val sigma: DBM[Double]) extends Serializable {
- private val sigmaInv2 = pinv(sigma) * -0.5
- private val U = math.pow(2.0 * math.Pi, -mu.length / 2.0) *
math.pow(det(sigma), -0.5)
-
+
+ private val (sigmaInv2, u) = calculateCovarianceConstants
+
/** Returns density of this multivariate Gaussian at given point, x */
def pdf(x: DBV[Double]): Double = {
val delta = x - mu
- val deltaTranspose = new Transpose(delta)
- U * math.exp(deltaTranspose * sigmaInv2 * delta)
+ u * math.exp(delta.t * sigmaInv2 * delta)
+ }
+
+ /*
+ * Calculate distribution dependent components used for the density
function:
+ * pdf(x) = (2*pi)^(-k/2) * det(sigma)^(-1/2) * exp( (-1/2) *
(x-mu).t * inv(sigma) * (x-mu) )
+ * where k is length of the mean vector.
+ *
+ * We here compute distribution-fixed parts
+ * (2*pi)^(-k/2) * det(sigma)^(-1/2)
+ * and
+ * (-1/2) * inv(sigma)
+ *
+ * Both the determinant and the inverse can be computed from the
singular value decomposition
+ * of sigma. Noting that covariance matrices are always symmetric and
positive semi-definite,
+ * we can use the eigendecomposition (breeze provides one specifically
for symmetric matrices,
+ * so I am making an assumption here that there is some efficiency gain).
+ *
+ * To guard against singular covariance matrices, this method computes
both the
+ * pseudo-determinant and the pseudo-inverse (Moore-Penrose). Singular
values are considered
+ * to be non-zero only if they exceed a tolerance based on machine
precision, matrix size, and
+ * relation to the maximum singular value (same tolerance used by, ie,
Octave).
+ */
+ private def calculateCovarianceConstants: (DBM[Double], Double) = {
+ val eigSym.EigSym(d, u) = eigSym(sigma) // sigma = u * diag(d) * u.t
+
+ // For numerical stability, values are considered to be non-zero only
if they exceed tol.
+ // This prevents any inverted value from exceeding (eps * n *
max(d))^-1
+ val tol = MLUtils.EPSILON * max(d) * d.length
+
+ // pseudo-determinant is product of all non-zero eigenvalues
+ val pdetSigma = (0 until d.length).map(i => if (d(i) > tol) d(i) else
1.0).reduce(_ * _)
--- End diff --
More concise:
```
val pdetSigma = d.activeValuesIterator.filter(_ > tol).foldLeft(1.0)(_ * _)
```
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