Github user mengxr commented on a diff in the pull request: https://github.com/apache/spark/pull/3871#discussion_r22486202 --- Diff: mllib/src/main/scala/org/apache/spark/mllib/stat/impl/MultivariateGaussian.scala --- @@ -17,23 +17,74 @@ 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. In + * the event that the covariance matrix is singular, the density will be computed in a + * reduced dimensional subspace under which the distribution is supported. + * (see [[http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case]]) + * + * @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) - + + /** + * Compute distribution dependent constants: + * sigmaInv2 = (-1/2) * inv(sigma) + * u = (2*pi)^(-k/2) * det(sigma)^(-1/2) + */ + private val (sigmaInv2: DBM[Double], u: Double) = 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. + * + * 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, e.g., 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 + + try { + // pseudo-determinant is product of all non-zero singular values + val pdetSigma = d.activeValuesIterator.filter(_ > tol).reduce(_ * _) + + // calculate pseudo-inverse by inverting all non-zero singular values + val pinvS = new DBV(d.map(v => if (v > tol) (1.0 / v) else 0.0).toArray) + val pinvSigma = u * diag(pinvS) * u.t --- End diff -- minor: We don't need to compute `pinvSigma` explicitly. ~~~ Sigma = U * D * U^T Sigma^+ = U * D^+ U^T = (D^{-1/2} U)^T (D^{-1/2} U) -0.5 * delta^T * Sigma^+ * delta = -0.5 \| D^{-1/2} U delta \|_2^2 ~~~ So we can scale the rows of `U` by `1/math.sqrt(d_i)`, and then update line 47.
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