Target audience: Intermediate

Estimated reading time: 20'

**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.

__Note__: The post requires some knowledge of data vectorization (numpy, datavec, ND4j..) as well as Scala programming language.

Vectorization

The

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

**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

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:

**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

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).*

1 2 3 4 5 6 | @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 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

The following implementation relies on the direct invocation of LAPACK library function

Note that the matrix is duplicated prior to the

Let's implement the multivariate Normal sampler with

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 '

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

Environment

Scala 2.13.2

JDK 1.8

ND4j 1.0.0-beta3

References

**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

The following implementation relies on the direct invocation of LAPACK library function

*potrf*. The LAPACK functionality are accessed through the BLAS wrapper*getBlasWrapper*.1 2 3 4 5 | 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.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | @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

ND4j 1.0.0-beta3

References

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

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