A fast matrix type for Swift

This is a basic matrix type that I wrote when I was playing with machine learning in Swift. It is by no means complete or finished.

The API is very loosely based on NumPy and Octave/MATLAB but more Swift-like.

Matrix uses the Accelerate.framework for most of its operations, so it should be pretty fast -- but no doubt there's lots of room for improvement.

Note: The Xcode project only includes unit tests, you can't run it. Press Cmd-U to perform the tests.

How to use it

For example, here's how you can use Matrix as part of the k-nearest neighbors algorithm:

// load your data set into matrix X, where each row represents one training
// example, and each column a feature
let X = Matrix(rows: 10000, columns: 200)

// load your test example into the row vector x
let x = Matrix(rows: 1, columns: 200)

// Calculate the distance between the test example and every training example
// and store this in a new column vector
let distances = (x.tile(X.rows) - X).pow(2).sumRows().sqrt()

The sqrt(), sumRows(), and pow() functions work on all the elements of the matrix. The - operator subtracts the matrices element-wise. This one-liner does the work of several loops, using accelerated CPU instructions from the BLAS, LAPACK, and vDSP libraries inside the Accelerate.framework.

Note: Matrix is a value type. Any operations return a new instance. That means it does not always optimally use memory. This should only be a problem if you're working with huge matrices and you have an algorithm that can work in-place, e.g. you want to subtract all elements in matrix B from matrix A and store the result back in A instead of a new matrix C.

Notes on the API

Notation is object-oriented, not mathematical

I decided to make all operations either member functions of Matrix or operators. There are no free functions that work on Matrix.

Even though the following reads more "mathematical",

sqrt(sumRows(pow(x.tile(X.rows) - X, 2)))

it requires you to unravel what happens "inside-out". Using member functions you can simply read from left-to-right:

(x.tile(X.rows) - X).pow(2).sumRows().sqrt()

Operators

The *, /, +, - operators on two matrices perform element-wise operations.

For example, A * B on two matrices A and B that have the same size, multiplies each element of matrix A with each element of matrix B. Like so:

    [ a b c ]        [ 1 2 3 ]            [ a*1 b*2 c*3 ]
A = [ d e f ]    B = [ 4 5 6 ]    A * B = [ d*4 e*5 f*6 ]
    [ g h i ]        [ 7 8 9 ]            [ g*7 h*8 i*9 ]

This is not the same as proper matrix-matrix (or matrix-vector) multiplication. For that, you have to use the special operator <*>. Likewise, </> is for dividing two matrices, i.e. multiplying one matrix with the inverse of another.

You can also use the *, /, +, - operators on a matrix and a row vector, in which case the operation happens on each of the columns of the matrix separately. And when you use a matrix and a column vector, the operation affects each of the rows of the matrix.

For example, a matrix and a row vector:

    [ a b c ]                             [ a*1 b*2 c*3 ]
X = [ d e f ]    v = [ 1 2 3 ]    X * v = [ d*1 e*2 f*3 ]
    [ g h i ]                             [ g*1 h*2 i*3 ]

and a matrix and a column vector:

    [ a b c ]        [ 1 ]                [ a*1 b*1 c*1 ]
X = [ d e f ]    v = [ 2 ]        X * v = [ d*2 e*2 f*2 ]
    [ g h i ]        [ 3 ]                [ g*3 h*3 i*3 ]

The TO-DO list

Accelerate:

• subscript(rows: Range<Int>) -> Matrix
• subscript(columns: Range<Int>) -> Matrix

Add new subscript:

• subscript(rows: Range<Int>, columns: Range<Int>) -> Matrix - this sets or gets a submatrix given by the two ranges
• subscript(columns: [Int]) -> Matrix - already have one for rows

Extend the functionality of:

• tile() currently only duplicates a row vector; in NumPy it can tile entire matrices in both directions.

Add functions for inserting/removing rows:

• insertRow(at:, repeatedValue:)
• insertRows(at:, repeatedValue:, count:)
• insertColumn(at:, repeatedValue:)
• insertColumns(at:, repeatedValue:, count:)
• remove(row: Int)
• remove(rows: Range<Int>)
• remove(column: Int)
• remove(columns: Range<Int>)

Other new operators:

• == operator on the elements of two matrices, writes 1.0 if true or 0.0 if false

Other ideas for improvements

Make it faster!

Make different versions and benchmark against one set of test data:

• Use ContiguousArray instead of Array?
• Use malloc or NSData to allocate memory instead of using Array?
• Use Float instead of Double?
• Maybe store the rows as columns and columns as rows? In a lot of machine learning stuff, computations are done on columns (where each column represents a feature). This may improve locality of the data.
• Use the new LinearAlgebra framework instead of LAPACK directly?

Some Accelerate.framework functions that might come in handy:

• catlas_dset() / vDSP_vfillD(): for initializing the "ones" matrix if we use malloc instead of Swift array.
• cblas_dnrm2() / vDSP_vdistD() / vDSP_vpythg(): Euclidian length of vector
• vDSP_normalizeD(): uses mean and std dev to normalize. This can also just calculate mean and stddev without normalizing, so maybe use this in the std() function.
• vDSP_svesqD(): sum of squares
• vDSP_mmovD(): copy submatrix

Slices

Make a MatrixSlice type, which works like ArraySlice. This is a view into a Matrix so you don't have to copy any data. Something like m[row: r] could return a MatrixSlice for just that row.

You can convert a slice into a copy of the data using Matrix(slice).

This also makes transpose() faster: it simply returns a MatrixSlice with a different "stride"; the actual data does not need to change.

See also how NumPy stores its arrays internally.

The end

Based on @mattt's Surge library.

Also check out this alternative matrix library for Swift: swix

MIT license