# 1. Tensor Overview¶

## 1.1. Synopsis¶

First, we will define a convenience alias:

typealias NativeStorage<Double> D


Next, we can create two vectors:

// construct vector with values
let v1 = Tensor<D>([1, 2, 3])

// construct Tensor with a specific size
let v2 = Tensor<D>(Extent(5, 3))

// construct a matrix with values
let v3 = Tensor<D>([[1, 2, 3], [4, 5, 6]])


STEM supports standard linear algebra operators:

// take the dot product (result is a scalar)
let s1 = v1⊙v2

// take the outer product (result is a matrix)
let m1 = v1⊗v2

let v4 = v1+v3

// multiply by a scalar
let v5 = 0.5*v1


STEM also supports advanced indexing (similar to Numpy and Matlab):

let v6 = v2[1..<4]
let m2 = m1[1..<4, 0..<2]


As STEM‘s name implies N-dimensional Tensors are supported. Both the Vector and Matrix classes are specializations of the Tensor class. These specializations allow for simpler construction methods as well as the use of accelerated libraries such as CBLAS and CUDA or OpenCL through function overloading.

## 1.2. Storage¶

All Tensor s have an associated Storage class that is responsible for the allocated memory. The two built-in Storage types are: NativeStorage and CBlasStorage. Other storage types (e.g. CUDA or OpenCL) can be added without requiring any rewrite of the main library. Because the Storage type determines which functions get called. If no methods have been specified for the Storage class, NativeStorage will be called by default.

The Storage protocol is defined as:

public protocol Storage {
associatedtype ElementType:NumericType

var size:Int { get }
var order:DimensionOrder { get }

init(size:Int)
init(array:[ElementType])
init(storage:Self)
init(storage:Self, copy:Bool)

subscript(index:Int) -> ElementType {get set}

// returns the order of dimensions to traverse
func calculateOrder(dims:Int) -> [Int]

// re-order list in order of dimensions to traverse
func calculateOrder(values:[Int]) -> [Int]
}


An implementation of Storage determines the allocation through the init methods, subscript determines how the storage gets indexed, and calculateStride allows the Storage to be iterated through in a sequential fashion.

The Tensor class frequently makes use of the generator IndexGenerator to iterate through the Storage class. This provides a convenient way to access all the elements without knowing the underyling memory allocation.

To do so, the Tensor class defined the method:

public func indices(order:DimensionOrder?=nil) -> GeneratorSequence<IndexGenerator> {
if let o = order {
return GeneratorSequence<IndexGenerator>(IndexGenerator(shape, order: o))
} else {
return GeneratorSequence<IndexGenerator>(IndexGenerator(shape, order: storage.order))
}
}


which can be used like:

func fill<StorageType:Storage>(tensor:Tensor<StorageType>, value:StorageType.ElementType) {
for i in tensor.indices() {
tensor.storage[i] = value
}
}


However, as mentioned previously, if an optimized version for a particular Tensor operation exists, you can write:

// This will be used if the Tensor's storage type is CBlasStorage for doubles,
// an alternative can be specified for Floats separately.
func fill(tensor:Tensor<CBlasStorage<Double>>, value:StorageType.ElementType) {
// call custom library
}

Storage Types
Type Description
NativeStorage Unaccelerated using row-major memory storage
CBlasStorage CBLAS acceleration using column-major storage
GPUStorage (Not Implemented) GPU acceleration using row-memory storage

## 1.3. Tensor Class¶

The Tensor class is parameterized by the Storage type, allowing instances of the class to maintain a pointer to the underlying memory. The Tensor class also has an instance of ViewType, which allows different views of the same memory to be constructed, and the array dimIndex, which determines the order that the dimensions in the Tensor are traversed. These features allow for multiple Tensor s to provide a different view to the same memory (e.g. a slice of a Tensor can be created by changing the ViewType instance, or a Tensor can be transposed by shuffling dimIndex).

Note

Throughout the documentation Tensor<S> indicates the parameterization of the Tensor class by Storage type S, and NumericType refers to S.NumericType (see section on Storage for details).

## 1.4. Tensor Construction¶

Tensor<S>(_ shape:Extent)

Constructs a tensor with the given shape.

Tensor<S>([NumericType], axis:Int)

Constructs a vector along the given axis

Tensor<S>(colvector:[NumericType])

Constructs a column vector (equivalent to Tensor<S>([NumericType], axis:0))

Tensor<S>(rowvector:[NumericType])

Constructs a row vector (equivalent to Tensor<S>([NumericType], axis:1))

Tensor<S>([[NumericType]])

Constructs a matrix

## 1.5. Indexing¶

STEM supports single indexing as well as slice indexing. Given a Tensor T:

To index element (i, j, k):

let value = T[i, j, k]
T[i, j, k] = value


To index the slices (if:il, jf:jl, kf:kl):

let T2 = T[if...il, jf...jl, kf...kl]
T[if...il, jf...jl, kf...kl] = T2


## 1.6. Views¶

Views in STEM are instances of Tensor that point to the same Storage as another Tensor but with different bounds and/or ordering of dimensions. Views are most commonly created whenever a slice indexing is used.

A copy of a view can be made by using the copy function.