Matrex

Fast matrix manipulation library for Elixir implemented in C native code with highly optimized CBLAS sgemm() used for matrix multiplication.

For example, vectorized linear regression is about 13 times faster, than Octave single threaded implementation.

It's also memory efficient, so you can work with large matrices, about billion of elements in size.

Based on matrix code from https://gitlab.com/sdwolfz/experimental/-/tree/master/exlearn

Benchmarks

Comparison with NumPy

2015 MacBook Pro, 2.2 GHz Core i7, 16 GB RAM

Operations are performed on 3000×3000 matrices filled with random numbers.

You can run benchmarks from the /bench folder with python numpy_bench.py and MIX_ENV=bench mix bench commands.

NumPy

benchmark         iterations	average time
logistic_cost()   1000         	1.23 ms/op
np.divide(A, B)   100           15.43 ms/op
np.add(A, B)      100           14.62 ms/op
sigmoid(A)        50            93.28 ms/op
np.dot(A, B)      10            196.57 ms/op

Matrex

benchmark     iterations   average time
logistic_cost()      1000  1.23 ms/op (on par)
divide(A, B)         200   7.32 ms/op (~ 2× faster)
add(A, B)            200   7.71 ms/op (~ 2× faster)
sigmoid(A)           50    71.47 ms/op (23% faster)
dot(A, B)            10    213.31 ms/op (8% slower)

Comparison with pure Elixir libraries

Slaughter of the innocents, actually.

2015 MacBook Pro, 2.2 GHz Core i7, 16 GB RAM

Dot product of 500×500 matrices

Dot product of 3×3 matrices

Transposing 1000x1000 matrix

Example

Complete example of Matrex library at work: Linear regression on MNIST digits (Jupyter notebook)

Visualization

Matrex implements Inspect protocol and looks nice in your console:

It can even draw a heatmap of your matrix in console! Here is an animation of logistic regression training with Matrex library and some matrix heatmaps:

Installation

The package can be installed by adding matrex to your list of dependencies in mix.exs:

def deps do
  [
    {:matrex, "~> 0.6"}
  ]
end

MacOS

Everything works out of the box, thanks to Accelerate framework. If you encounter a compilation error

native/src/matrix_dot.c:5:10: fatal error: 'cblas.h' file not found

then make sure the XCode command-line tools are installed (xcode-select --install). If the error still not resolved, for MacOS Mojave, run open /Library/Developer/CommandLineTools/Packages/macOS_SDK_headers_for_macOS_10.14.pkg to restore /usr/include and /usr/lib.

On MacOS 10.15 this error can be solved with

export CPATH=/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/

or with

C_INCLUDE_PATH=/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/System/Library/Frameworks/Accelerate.framework/Frameworks/vecLib.framework/Headers mix compile

Ubuntu

You need to install scientific libraries for this package to compile:

> sudo apt-get install build-essential erlang-dev libatlas-base-dev

Windows

It will definitely work on Windows, but we need a makefile and installation instruction. Please, contribute.

Choosing which BLAS to use

With the help of MATREX_BLAS environment variable you can choose which BLAS library to link with. It can take values blas (the default), atlas, openblas or noblas.

The last option means that you compile C code without any external dependencies, so, it should work anywhere with a C compiler place:

$ mix clean
$ MATREX_BLAS=noblas mix compile

Access behaviour

Access behaviour is partly implemented for Matrex, so you can do:

    iex> m = Matrex.magic(3)
    #Matrex[3×3]
    ┌                         ┐
    │     8.0     1.0     6.0 │
    │     3.0     5.0     7.0 │
    │     4.0     9.0     2.0 │
    └                         ┘
    iex> m[2][3]
    7.0

Or even:

    iex> m[1..2]
    #Matrex[2×3]
    ┌                         ┐
    │     8.0     1.0     6.0 │
    │     3.0     5.0     7.0 │
    └                         ┘

There are also several shortcuts for getting dimensions of matrix:

    iex> m[:rows]
    3

    iex> m[:size]
    {3, 3}

calculating maximum value of the whole matrix:

    iex> m[:max]
    9.0

or just one of it's rows:

    iex> m[2][:max]
    7.0

calculating one-based index of the maximum element for the whole matrix:

    iex> m[:argmax]
    8

and a row:

    iex> m[2][:argmax]
    3

Math operators overloading

Matrex.Operators module redefines Kernel math operators (+, -, *, / <|>) and defines some convenience functions, so you can write calculations code in more natural way.

It should be used with great caution. We suggest using it only inside specific functions and only for increased readability, because using Matrex module functions, especially ones which do two or more operations at one call, are 2-3 times faster.

Usage example

    def lr_cost_fun_ops(%Matrex{} = theta, { %Matrex{} = x, %Matrex{} = y, lambda } = _params)
        when is_number(lambda) do
      # Turn off original operators
      import Kernel, except: [-: 1, +: 2, -: 2, *: 2, /: 2, <|>: 2]
      import Matrex.Operators
      import Matrex

      m = y[:rows]

      h = sigmoid(x * theta)
      l = ones(size(theta)) |> set(1, 1, 0.0)

      j = (-t(y) * log(h) - t(1 - y) * log(1 - h) + lambda / 2 * t(l) * pow2(theta)) / m

      grad = (t(x) * (h - y) + (theta <|> l) * lambda) / m

      {scalar(j), grad}
    end

The same function, coded with module methods calls (2.5 times faster):

    def lr_cost_fun(%Matrex{} = theta, { %Matrex{} = x, %Matrex{} = y, lambda } = _params)
        when is_number(lambda) do
      m = y[:rows]

      h = Matrex.dot_and_apply(x, theta, :sigmoid)
      l = Matrex.ones(theta[:rows], theta[:cols]) |> Matrex.set(1, 1, 0)

      regularization =
        Matrex.dot_tn(l, Matrex.square(theta))
        |> Matrex.scalar()
        |> Kernel.*(lambda / (2 * m))

      j =
        y
        |> Matrex.dot_tn(Matrex.apply(h, :log), -1)
        |> Matrex.subtract(
          Matrex.dot_tn(
            Matrex.subtract(1, y),
            Matrex.apply(Matrex.subtract(1, h), :log)
          )
        )
        |> Matrex.scalar()
        |> (fn
              :nan -> :nan
              x -> x / m + regularization
            end).()

      grad =
        x
        |> Matrex.dot_tn(Matrex.subtract(h, y))
        |> Matrex.add(Matrex.multiply(theta, l), 1.0, lambda)
        |> Matrex.divide(m)

      {j, grad}
    end

Enumerable protocol

Matrex implements Enumerable, so, all kinds of Enum functions are applicable:

    iex> Enum.member?(m, 2.0)
    true

    iex> Enum.count(m)
    9

    iex> Enum.sum(m)
    45

For functions, that exist both in Enum and in Matrex it's preferred to use Matrex version, beacuse it's usually much, much faster. I.e., for 1 000 x 1 000 matrix Matrex.sum/1 and Matrex.to_list/1 are 438 and 41 times faster, respectively, than their Enum counterparts.

Saving and loading matrix

You can save/load matrix with native binary file format (extra fast) and CSV (slow, especially on large matrices).

Matrex CSV format is compatible with GNU Octave CSV output, so you can use it to exchange data between two systems.

Example

    iex> Matrex.random(5) |> Matrex.save("rand.mtx")
    :ok
    iex> Matrex.load("rand.mtx")
    #Matrex[5×5]
    ┌                                         ┐
    │ 0.05624 0.78819 0.29995 0.25654 0.94082 │
    │ 0.50225 0.22923 0.31941  0.3329 0.78058 │
    │ 0.81769 0.66448 0.97414 0.08146 0.21654 │
    │ 0.33411 0.59648 0.24786 0.27596 0.09082 │
    │ 0.18673 0.18699 0.79753 0.08101 0.47516 │
    └                                         ┘
    iex> Matrex.magic(5) |> Matrex.divide(Matrex.eye(5)) |> Matrex.save("nan.csv")
    :ok
    iex> Matrex.load("nan.csv")
    #Matrex[5×5]
    ┌                                         ┐
    │    16.0     ∞       ∞       ∞       ∞   │
    │     ∞       4.0     ∞       ∞       ∞   │
    │     ∞       ∞      12.0     ∞       ∞   │
    │     ∞       ∞       ∞      25.0     ∞   │
    │     ∞       ∞       ∞       ∞       8.0 │
    └                                         ┘

NaN and Infinity

Float special values, like :nan and :inf live well inside matrices, can be loaded from and saved to files. But when getting them into Elixir they are transferred to :nan,:inf and :neg_inf atoms, because BEAM does not accept special values as valid floats.

    iex> m = Matrex.eye(3)
    #Matrex[3×3]
    ┌                         ┐
    │     1.0     0.0     0.0 │
    │     0.0     1.0     0.0 │
    │     0.0     0.0     1.0 │
    └                         ┘

    iex> n = Matrex.divide(m, Matrex.zeros(3))
    #Matrex[3×3]
    ┌                         ┐
    │     ∞      NaN     NaN  │
    │    NaN      ∞      NaN  │
    │    NaN     NaN      ∞   │
    └                         ┘

    iex> n[1][1]
    :inf

    iex> n[1][2]
    :nan

Creating Matrex logo from MNIST letters

    iex(166)> matrex_logo = \
    ...(166)> "../emnist/emnist-letters-test-images-idx3-ubyte" \
    ...(166)> |> Matrex.load(:idx) \
    ...(166)> |> Access.get(9601..10200) \
    ...(166)> |> Matrex.list_of_rows() \
    ...(166)> |> Enum.reduce(fn x, sum -> add(x, sum) end) \
    ...(166)> |> Matrex.reshape(28, 28) \
    ...(166)> |> Matrex.transpose() \
    ...(166)> |> Matrex.resize(2) \
    ...(166)> |> Matrex.heatmap(:color24bit)