multilinear-math

Swift library for multidimensional data, tensor operations and machine learning on OS X. For additional comments and documentation, please take a look at the Wiki.

Already implemented:

Swift wrappers of many important functions from the Accelerate framework and LAPACK (vector summation, addition, substraction, matrix and elementwise multiplication, division, matrix inverse, pseudo inverse, eigendecomposition, singular value decomposition...)
MultidimensionData protocol for elegant handling of multidimensional data of any kind
Clear, compact and powerful syntax for mathematical operations on tensors
Principal component analysis
Multilinear subspace learning algorithms for dimensionality reduction
Linear and logistic regression
Stochastic gradient descent
Feedforward neural networks
Sigmoid, ReLU, Softplus activation functions
Easy regularizations

Tensor reading, writing, slicing

Create data tensor:

var a = Tensor<Float>(modeSizes: [3, 3, 3], repeatedValue: 0)

Read and write single values:

let b: Float = a[1, 2, 0]
a[2, 0, 1] = 3.14

Read and write a tensor slice

let c: Tensor<Float> = a[1..<3, all, [0]]
a[1...1, [0, 2], all] = Tensor<Float>(modeSizes: [2, 3], values: [1, 2, 3, 4, 5, 6])

modes of size 1 will be trimmed

Einstein notation

Modes with same symbolic index will be summed over. Simple matrix multiplication:

var m = Tensor<Float>(modeSizes: [4, 6], repeatedValue: 1)
var n = Tensor<Float>(modeSizes: [6, 5], repeatedValue: 2)
let matrixProduct = m[.i, .j] * n[.j, .k]

Geodesic deviation:

var tangentVector = Tensor<Float>(modeSizes: [4], values: [0.3, 1.7, 0.2, 0.5])
tangentVector.isCartesian = false

var deviationVector = Tensor<Float>(modeSizes: [4], values: [0.1, 0.9, 0.4, 1.2])
deviationVector.isCartesian = false

var riemannianTensor = Tensor<Float>(diagonalWithModeSizes: [4, 4, 4, 4])
riemannianTensor.isCartesian = false
riemannianTensor.variances = [.contravariant, .covariant, .covariant, .covariant]

let relativeAcceleration = riemannianTensor[.μ, .ν, .ρ, .σ] * tangentVector[.ν] * tangentVector[.ρ] * deviationTensor[.σ]

Neural networks

Setting up and training a simple feedforward neural net:

var estimator =  NeuralNet(layerSizes: [28*28, 40, 10])
estimator.layers[0].activationFunction = ReLU(secondarySlope: 0.01) 
estimator.layers[1].activationFunction = ReLU(secondarySlope: 0.01)

var neuralNetCost = SquaredErrorCost(forEstimator: estimator)

stochasticGradientDescent(neuralNetCost, inputs: trainingData[.a, .b], targets: trainingLabels[.a, .c], updateRate: 0.1, minibatchSize: 50, validationCallback: ({ (epoch, estimator) -> (Bool) in
    if(epoch >= 30) {return true} 
    else {return false}
}))

documentation

Multilinear subspace learning

Extended PCA algorithms to work with tensors with arbitrary mode count

multilinear principal component analysis (MPCA)
uncorrelated multilinear principal component analysis (UMPCA)
(Lu, Plataniotis, Venetsanopoulos)

Installation

To use this framework in an OSX XCode project:

clone this repository
open your project in XCode
drag and drop MultilinearMath.xcodeproj into the project navigator of your project
select the .xcodeproj file of your project in the navigator
go to the "General" tab and add MultilinearMath.framework to the Linked Frameworks and Libraries
go to the "Build Settings" tab and set "Embedded Content Contains Swift Code" to YES
import MultilinearMath in your Swift files

If possible, whole module optimization should be used for very significant performance gains.

License

vincentherrmann/multilinear-math

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