Sunday, June 9, 2019

Multivariate Normal sampling with Scala and ND4j

This post describes the implementation of the multivariate normal sampling using ND4j. The multi-variate normal sampling function is used in various machine learning techniques such as Markov chains Monte Carlo (MCMC) or contextual multi-armed bandit.
The implementation of the multivariate normal relies on data vectorization, technique well-known to Python developers and data scientists alike, that we introduce in the first section.

Vectorization
Python Numpy is a well-known and reliable vectorized linear algebra library which is a foundation of scientific (SciPy) and machine learning (Sciktlearn) libraries. No serious data scientific projects can reasonably succeeds with the power and efficiency of numpy. 
The vectorization of datasets is the main reason behind the performance of machine learning models (training and prediction) build in Python.

Is there a similar linear algebra library, supporting vectorization, available to Scala and Spark developers? Yes, ND4j


ND4j, BLAS and LAPACK
ND4j library replicates the functionality of numpy for Java developers. ND4j can be downloaded as an independent library or as component of the deep learning library, deeplearning4j. It leverages the BLAS and LAPACK libraries.
From a Java developer perspective, the data represented by an NDArray is stored outside of the Java Virtual Machine. This design has the following benefits:
  • Avoid taxing the Garbage Collector
  • Interoperability with high-performance BLAS libraries such as OpenBlas
  • Allow number of array elements exceeds Int.MaxValue
The BLAS (Basic Linear Algebra Subprograms) are functions performing basic vector and matrix operations. The library is divided in 3 levels:
  • Level 1 BLAS perform scalar, vector and vector-vector operations,
  • Level 2 BLAS perform matrix-vector operations
  • Level 3 BLAS perform matrix-matrix operations. 
OpenBLAS is an optimized Basic Linear Algebra Subprograms (BLAS) library based on GotoBLAS2,  a C-library of linear algebra supporting a large variety of micro-processor. Its usage is governed by the BSD license.
LAPACK are Fortran routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems and matrix factorizations.

Implicit ND4j array conversion
The first step is to create a implicit conversation between ND4j and Scala data types.  
The following code convert an array of double into a INDArray using org.nd4j.linalg.factory.Nd4j java class and its constructor create(double[] values, int[] shape)
  • In case of a vector, the shape is defined in scala as  Array[Int](size_vector)
  • In case of a matrix, the shape is defined as Array[Int](numRows, numCols). 
The following snippet implement a very simple conversion from a Scala array to a INDArray

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@throws(clazz = classOf[IllegalArgumentException])
implicit def double2NDArray(values: Array[Double]): INDArray = {
  require(values.nonEmpty, "ERROR: ND4, conversion ...")

  Nd4j.create(values, Array[Int](1, values.length))
}

Multivariate Normal distribution implementation
The sampling of a multivariate normal distribution is defined by the formula 
\[\mathbb{N}(\mu, \Sigma )=\frac{1}{\sqrt{(2\pi)^n \left | \Sigma \right |}} e^{\frac{1}{2}(x-\mu)^T {\Sigma '}^{-1}(x-\mu)}\] A generalized definition adds a very small random perturbation factor r |r| <= 1 on the variance value (diagonal of the covariance matrix) \[\Sigma ' = \Sigma + r.I\] Sigma is the covariance matrix and the mu is the mean value. 
The computation of the multivariate normal sampling can be approximated by the Cholesky decomposition. In a nutshell, the Cholesky algorithm decompose a positive-definite matrix into a product of two matrix
  • lower triangle matrix
  • transposition of its conjugate
It serves the same purpose as the ubiquitous LU decomposition with less computation. \[\mathbb{N}(\mu, \Sigma) \sim \mu + Chol(\Sigma).Z^T\] where \[L=Chol(\Sigma)\] and \[L.L^T=\Sigma\]. The vector Z is a multivariate normal noise \[Z= \{ z_i | z_i=N(0, 1)\}\]
The following implementation relies on the direct invocation of LAPACK library function potrf. The LAPACK functionality are accessed through the BLAS wrapper getBlasWrapper.

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 final def choleskyDecomposition(matrix: INDArray): INDArray = {
   val matrixDup = matrix.dup
   Nd4j.getBlasWrapper.lapack.potrf(matrixDup, false)
   matrixDup
 }

Note that the matrix is duplicated prior to the LAPACK function call as we do not want to alter the input matrix. 
Let's implement the multivariate Normal sampler with perturbation using the Cholesky decomposition.


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@throws(clazz = classOf[IllegalArgumentException])
@throws(clazz = classOf[IllegalStateException])
final def multiVariateNormalSampling(
   mean: INDArray,
   cov: INDArray,
   perturbation: Double = 0.0): INDArray = {
 import scala.util.Random
 require(cov.size(0) == mean.length, s"Sigma shape ${cov.size(0)} should be mean size ${mean.length}")
 require(cov.size(1) == mean.length, s"Sigma shape ${cov.size(1)} should be ${mean.length}")

 try {
  // Add a perturbation epsilon value to the covariance matrix
  // if it is defined
  val perturbMatrix =
   if(perturbation > 0.0)
    cov.add( squareIdentityMatrix(cov.size(0), 
             perturbation*(Random.nextDouble-0.5)))
   else
    cov

  // Apply the Cholesky decomposition
  val perturbed: INDArray = choleskyDecomposition(perturbMatrix)
   // Instantiate a normal distribution
  val normalPdf = new NormalDistribution(
       new DefaultRandom, 
       0.0, 
       1.0)

  val normalDensity = Array.fill(mean.size(0))(normalPdf.sample)
  val normalNDArray: INDArray = normalDensity

   // This is an implementation of the Dot product
  val normalCov = perturbed.mmul(normalNDArray.transpose)
   // Add the normalized perturbed covariance to the mean value
  mean.add(normalCov)
 }
 catch {
  case e: org.nd4j.linalg.api.blas.BlasException =>
   throw new IllegalStateException(s"multiVariateNormalSampling: ${e.toString}")
  case e: org.apache.commons.math3.exception.NotStrictlyPositiveException =>
   throw new IllegalStateException(s"multiVariateNormalSampling: ${e.toString}")
  case e: Throwable =>
   throw new IllegalStateException(s"multiVariateNormalSampling: ${e.toString}")
 }
}

Let's look at the full implementation of the multi-variate normal sampling.
The first step validates the shape of the mean and covariance input matrices [line 8, 9]. As mentioned earlier, the generalized normal distribution introduces an optional random perturbation of small magnitude (~1e-4) [line 14-17] that is useful for application that requires some stochastic
The 'perturbed' covariance matrix is factorized using the Cholesky decomposition [line 22]. The normal probability density function (mean 0.0 and standard deviation 1.0) is used to generate random values [line 24-30] which is then applied to the covariance matrix [line 33].
The normal randomized variance is added to the vector of mean values [line 35].

For the sake of convenience, the multivariate normal density function uses the Apache Math common 3.5 API [line 24].

Environment
Scala 2.13.2
JDK 1.8

References
Scala 2.13
ND4j
Deep learning 4j
Cholesky decomposition
Approximation multivariate Normal sampling
Apache common math

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