Owl is an OCaml numerical library. It supports N-dimensional arrays, both dense and sparse matrix operations, linear algebra, regressions, fast Fourier transforms, and many advanced mathematical and statistical functions (such as Markov chain Monte Carlo methods). Recently, Owl has implemented algorithmic differentiation which essentially makes developing machine learning and neural network algorithms trivial.

The full API documentation is here (maybe outdated):

• on mirage.io
• on cam.ac.uk

The series of tutorials is here (more is coming):

• Tutorial 1: Basic Data Types
• Tutorial 2: N-dimensional Array
• Tutorial 3: Matrix Manipulation
• Tutorial 4: How to Plot in Owl?
• Tutorial 5: Metric Systems
• Tutorial 6: Indexing and Slicing
• Tutorial 7: Operators and Ext Module
• Tutorial 8: Algorithmic Differentiation

Some simple evaluations can be found as follows [Ndarray]. The roadmap and future plan of Owl can be found [Here]. I would love to hear from you, and please let me know your comments and suggestions to improve Owl.

Email Me or message me on: Twitter, Google+, Facebook, Blogger, LinkedIn

Owl requires OCaml 4.04.0. The installation is rather trivial. First, you need to clone the repository.

git clone [email protected]:ryanrhymes/owl.git

Then you need to install all the dependencies.

opam install ctypes dolog eigen gsl oasis plplot

Next, you can compile and install the module with the following command.

make oasis
make && make install

Owl is well integrated with utop. Now you can start utop and continue this tutorial to do some experiments. If you want utop to automatically load Owl for you, you can also edit .ocamlinit file in your home folder by adding the following lines. (Note that the library name is owl with lowercase o.)

#require "owl"

If you are too lazy to do any labour work, here is a docker image to let you try Owl without dealing with aforementioned installation and configuration steps. The docker image is automatically build from the master branch whenever there are new commits. You can check the building history on Docker Hub.

Just pull the image, start a container, then play with it in utop. The latest source code is saved in /root/owl directory.

docker pull ryanrhymes/owl
docker run -t -i ryanrhymes/owl

Owl currently has the following core modules and their names all start with Owl_ to avoid name conflicts, e.g., Owl_dense, Owl_sparse, Owl_maths, Owl_stats, Owl_const, Owl_fft, Owl_plot and etc. After utop successfully loads Owl library, you can access the module functions using aforementioned names.

However, a more convenient way is to use Owl module as an entry point which provides aliases of the core module names for easy access, e.g., Owl.Dense is the same as Owl_dense, and Owl.Regression is the same as Owl_regression. Given no name conflicts, you can simply open the whole Owl module for convenience as I will do in the rest of this tutorial.

open Owl;;

Dense.Matrix module supports dense matrix operations while Sparse.Matrix module supports sparse ones. There are five submodules in Dense.Matrix:

• Dense.Matrix.S module supports single precision float numbers float32;
• Dense.Matrix.D module supports double precision float numbers float64;
• Dense.Matrix.C module supports single precision complex numbers complex32;
• Dense.Matrix.Z module supports double precision complex numbers complex64;
• Dense.Matrix.Generic module supports all aforementioned number types via GADT.

To start, we can use Dense.Matrix.D.uniform_int to create a 5x5 random dense matrix.

let x = Dense.Matrix.D.uniform_int 5 5;;

You should see the following output in utop.

   C0 C1 C2 C3 C4
R0 25  2 77 85 72
R1 71 92 98 87 53
R2 35 29 82 65 20
R3  2 29 66 42 12
R4 99 72 78 30 11
val x : Owl_dense_matrix_d.mat =

To save some typing efforts, we have made Mat as an alias of Dense.Matrix.D by assuming 64-bit float numbers are commonly used. Therefore, we can use Mat directly after open Owl instead of using Dense.Matrix.D. Similarly, there are also aliases for 64-bit float vectors and ndarrays (i.e., Vec and Arr) but we will talk about them later. Mat module also provides other functions to create various matrices, e.g., as below.

let x = Mat.eye 5;;             (* identity matrix *)
let x = Mat.zeros 5 5;;         (* all elements are zeros *)
let x = Mat.ones 5 5;;          (* all elements are ones *)
let x = Mat.uniform 5 5;;       (* random matrix of uniform distribution *)
let x = Mat.gaussian 5 5;;      (* random matrix of gaussian distribution *)
...

Combined with Stats module, you can also create any matrices of many distributions. E.g., the following code first creates an empty dense matrix, then initialise the elements with Bernoulli distribution. Test it in utop, you should get a dense matrix where half of the elements are zeros.

let x = Mat.empty 8 8 |> Mat.map (fun _ -> Stats.Rnd.bernoulli 0.5 |> float_of_int);;

Or create a matrix where the elements follow Laplace distribution.

let x = Mat.empty 8 8 |> Mat.map (fun _ -> Stats.Rnd.laplace 0.2);;

With Dense module, you can also generate linearly spaced interval and meshgrids, e.g.,

let x = Mat.linspace 0. 5. 6;;

which will return a 1x5 row vector as below

   C0  C1  C2  C3  C4  C5
R0  0   1   2   3   4   5
val x : Owl_dense_matrix_d.mat =

The created matrices can be casted into other number types easily. For example the following code casts a float32 matrix x into complex64 matrix y.

let x = Dense.Matrix.S.uniform 3 3;;
let y = Dense.Matrix.Generic.cast_s2z x;;

Matrices can be saved to and loaded from a file.

Mat.save x "matrix_01.data";;  (* save the matrix to a file *)
Mat.load "matrix_01.data";;    (* load the matrix from a file *)

Both Dense.Matrix and Sparse.Matrix modules provide a wide range of operations to access the elements, rows, and columns of a matrix. You can refer to the full document in Dense.Matrix.Generic. Here we just gave some simple examples briefly.

You can use Mat.set and Mat.get to manipulate individual element.

Mat.set x 0 1 2.5;;
Mat.get x 0 1;;

Equivalently, there are shorthands for Mat.get and Mat.set.

x.{0,1} <- 2.5;;  (* Mat.set x 0 1 2.5 *)
x.{0,1};;         (* Mat.get x 0 1 *)

We can use Mat.row and Mat.col to retrieve a specific row or column of a matrix, or use Mat.rows and Mat.cols to retrieve multiple of them.

Mat.row x 5;;            (* retrieve the fifth row *)
Mat.cols x [|1;3;2|];;   (* retrieve the column 1, 3, and 2 *)

E.g., the following code generates a random matrix, then scales up each element by a factor of 10 using Mat.map function.

let x = Mat.(uniform 6 6 |> map (fun x -> x *. 10.));;

We can iterate a matrix row by row, or column by column. The following code calculates the sum of each row by calling Mat.map_rows function.

let x = Mat.(uniform 6 6 |> map_rows sum);;

We can fold elements by calling Mat.fold, fold rows by calling Mat.fold_rows. Similarly, there are also functions for filter operations. The following code filters out the elements not greater than 0.1 in x.

Mat.filter ((>) 0.1) x;;    (* not greater than 0.1 in x *)

We can also do something more complicated, e.g., by filtering out the rows whose summation is greater than 3.

Mat.filter_rows (fun r -> Mat.sum r > 3.) x;;

Shuffle the rows and columns, or draw some of them from a matrix.

Mat.shuffle_rows x;;     (* shuffle the rows in x *)
Mat.draw_cols x 3;;      (* draw 3 columns from x with replacement *)
...

Practically, Sparse.Matrix module provides a subset of the similar operations for sparse matrices. In addition, Sparse.Matrix module also has extra functions such as only iterating non-zero elements Sparse.Matrix.Generic.iter_nz, and etc. Please read the full documentation for Sparse.Matrix.Generic for details.

Simple matrix mathematics like add, sub, multiplication, and division are included in Dense module. Moreover, there are predefined shorthands for such operations. E.g., the following code creates two random matrices then compare which is greater.

let x = Mat.uniform 6 6;;
let y = Mat.uniform 6 6;;
Mat.(x > y);;                (* is x greater than y? return boolean *)
Mat.(x = y);;                (* is x equal to y? *)
Mat.(x <. y);;               (* is x smaller than y? return 0/1 matrix *)
...

Owl natively supports broadcast operation similar to other numerical libraries. Some basic math operations includes:

Mat.(x + y);;                (* addition of two matrices *)
Mat.(x * y);;                (* element-wise multiplication *)
Mat.(x *@ y);;               (* matrix multiplication of two matrices *)
Mat.(x +$ 2.);;              (* add a scalar to all elements in x *)
...

Apply various functions in Maths module to every element in x

Mat.(Maths.atanh @@ x);;        (* apply atanh function *)
Mat.(Maths.airy_Ai @@ x);;      (* apply Airy function *)
...

However, it is worth pointing out that Mat already implements many useful math functions. These functions are vectorised and are much faster than the example above which actually calls Mat.map for transformation.

Mat.sin x;;         (* call sine function *)
Mat.erfc x;;        (* call erfc function *)
Mat.round x;;       (* call round function *)
Mat.signum x;;      (* call signum function *)
Mat.sigmoid x;;     (* apply sigmoid function *)
...

Concatenate two matrices, vertically or horizontally by

Mat.(x @= y);;                (* equivalent to Mat.concat_vertical *)
Mat.(x @|| y);;               (* equivalent to Mat.concat_horizontal *)

More advanced linear algebra operations such as svd, qr, and cholesky decomposition are included in Linalg module.

let u,s,v = Linalg.svd x;;   (* singular value decomposition *)
let q,r = Linalg.qr x;;      (* QR decomposition *)
let l = Linalg.cholesky x;;  (* cholesky decomposition *)
...

Regression module currently includes linear, exponential, nonlinear, ols, ridge, lasso, svm, and etc. Most of them are based on a stochastic gradient descent algorithm implemented in Optimise module.

In the following, let's use an example to illustrate the simplest linear regression in Regression module. First, let's generate the measurement x which is a 1000 x 3 matrix. Each row of x is an independent measurement.

let x = Mat.uniform 1000 3;;

Next let's define the parameter of a linear model, namely p, a 3 x 1 matrix.

let p = Mat.of_array [|0.2;0.4;0.8|] 3 1;;

Then we generate the observations y from x and p by

let y = Mat.(x *@ p);;

Now, assume we only know x and y, how can we fit x and y into a linear model? It is very simple.

let p' = Regression.linear x y;;

From utop, you can see that p' equals [|0.2; 0.4; 0.8|] which is exactly the same as p. For other regression such as lasso and svm, the operation is more or less the same, please read Owl document for details.

There is another separate Tutorial on Plotting in Owl. Herein, let's use an example to briefly show how to plot the result using Plot module. We first generate two mesh grids then apply sine function to them by using the operations introduced before.

let x, y = Mat.meshgrid (-2.5) 2.5 (-2.5) 2.5 100 100 in
let z = Mat.(sin ((x * x) + (y * y))) in
Plot.mesh x y z;;

No matter what plot terminal you use, you should end up with a figure as below.

Besides Plot.mesh, there are several other basic plotting functions in Plot. Even though the module is still immature and under active development, it can already do some fairly complicated plots with minimal coding efforts. E.g., the following code will generate a 2 x 2 subplot.

let f p i = match i with
  | 0 -> Stats.Rnd.gaussian ~sigma:0.5 () +. p.(1)
  | _ -> Stats.Rnd.gaussian ~sigma:0.1 () *. p.(0)
in
let y = Stats.gibbs_sampling f [|0.1;0.1|] 5_000 |> Mat.of_arrays in
let h = Plot.create ~m:2 ~n:2 "test_plot_04.png" in
let _ = Plot.set_background_color h 255 255 255 in
let _ = Plot.subplot h 0 0 in
let _ = Plot.set_title h "Bivariate model" in
let _ = Plot.scatter ~h (Mat.col y 0) (Mat.col y 1) in
let _ = Plot.subplot h 0 1 in
let _ = Plot.set_title h "Distribution of y" in
let _ = Plot.set_xlabel h "y" in
let _ = Plot.set_ylabel h "Frequency" in
let _ = Plot.histogram ~h ~bin:50 (Mat.col y 1) in
let _ = Plot.subplot h 1 0 in
let _ = Plot.set_title h "Distribution of x" in
let _ = Plot.set_ylabel h "Frequency" in
let _ = Plot.histogram ~h ~bin:50 (Mat.col y 0) in
let _ = Plot.subplot h 1 1 in
let _ = Plot.set_foreground_color h 51  102 255 in
let _ = Plot.set_title h "Sine function" in
let _ = Plot.plot_fun ~h ~line_style:2 Maths.sin 0. 28. in
let _ = Plot.autocorr ~h (Mat.sequential 1 28) in
Plot.output h;;

The end result is as follows. You probably have already grasped the idea of how to plot in Owl. But I promise to write another separate post to introduce plotting in more details.

There are a lot of basic and advanced mathematical and statistical functions in Maths and Stats modules. Most of them are interfaced to Gsl directly, so you may want to read GSL Manual carefully before using the module. In the future, Owl will also supports other math library as optional backend in case you need different licence.

Stats has three submodules: Stats.Rnd for random numbers, Stats.Pdf for probability dense functions, and Stats.Cdf for cumulative distribution functions. In addition, I have implemented extra functions such as two ranking correlations: Stats.kendall_tau and Stats.spearman_rho); two MCMC (Markov Chain Monte Carlo) functions in Stats module: Metropolis-Hastings (Stats.metropolis_hastings) and Gibbs sampling (Stats.gibbs_sampling) algorithms.

E.g., the following code first defines a probability density function f for a mixture Gaussian model. Then we use Stats.metropolis_hastings to draw 100_000 samples based on the given pdf f, and the initial point is 0.1. In the end, we call Plot.histogram to plot the distribution of the samples, from which we can clearly see they are from a mixture Gaussian model.

let f p = Stats.Pdf.((gaussian p.(0) 0.5) +. (gaussian (p.(0) -. 3.5) 1.)) in
let y = Stats.metropolis_hastings f [|0.1|] 100_000 |>  Mat.of_arrays in
Plot.histogram ~bin:100 y;;

The histogram below shows the distribution of the samples.

Here is another example using Stats.gibbs_sampling to sample a bivariate distribution. Gibbs sampling requires the full conditional probability function so we defined its corresponding random number generator in f p i where p is the parameter vector and i indicates which parameter to sample.

let f p i = match i with
  | 0 -> Stats.Rnd.gaussian ~sigma:0.5 () +. p.(1)
  | _ -> Stats.Rnd.gaussian ~sigma:0.1 () *. p.(0)
in
let y = Stats.gibbs_sampling f [|0.1;0.1|] 5_000 |> Mat.of_arrays in
Plot.scatter (Mat.col y 0) (Mat.col y 1);;

We take 5000 samples from the defined distribution and plot them as a scatter plot, as below.

The future plan is to embed a small PPL (Probabilistic Programming Language) in Stats module.

Owl has a very powerful module to manipulate dense N-dimensional arrays, i.e., Dense.Ndarray. Ndarray is very similar to the corresponding modules in Numpy and Julia. For sparse N-dimensional arrays, you can use Sparse.Ndarray which provides a similar set of APIs as aforementioned Ndarray. Here is an initial evaluation on the performance of Ndarray.

Similar to Matrix module, Ndarray also has five submodules S (for float32), D (for float32), C (for complex32), Z (for complex64), and Generic (for all types) to handle different number types. There is an alias in Owl for double precision float ndarray (i.e., Dense.Ndarray.D) which is Arr. Ndarray also natively supports broadcast operations

In the following, I will present a couple of examples using Dense.Ndarray module. First, we can create empty ndarrays of shape [|3;4;5|].

let x0 = Dense.Ndarray.S.empty [|3;4;5|];;
let x1 = Dense.Ndarray.D.empty [|3;4;5|];;
let x2 = Dense.Ndarray.C.empty [|3;4;5|];;
let x3 = Dense.Ndarray.Z.empty [|3;4;5|];;

You can also assign the initial values to the elements, generate a zero/one ndarray, or even a random ndarray.

Dense.Ndarray.C.zeros [|3;4;5|];;
Dense.Ndarray.D.ones [|3;4;5|];;
Dense.Ndarray.S.create [|3;4;5|] 1.5;;
Dense.Ndarray.Z.create [|3;4;5|] Complex.({im=1.5; re=2.5});;
Dense.Ndarray.D.uniform [|3;4;5|];;

With these created ndarray, you can do some math operation as below. Now, let's use shortcut Arr module to make examples.

let x = Arr.uniform [|3;4;5|];;
let y = Arr.uniform [|3;4;5|];;
let z = Arr.add x y;;
Arr.print z;;

Owl supports many math operations and these operations have been well vectorised so they are very fast.

Arr.sin x;;
Arr.tan x;;
Arr.exp x;;
Arr.log x;;
Arr.min x;;
Arr.add_scalar x 2.;;
Arr.mul_scalar x 2.;;
...

Examining elements and comparing two ndarrays are also very easy.

Arr.is_zero x;;
Arr.is_positive x;;
Arr.is_nonnegative x;;
...
Arr.equal x y;;
Arr.greater x y;;
Arr.less_equal x y;;
...

You can certainly plugin your own functions to check each elements.

Arr.exists ((>) 2.) x;;
Arr.not_exists ((<) 2.) x;;
Arr.for_all ((=) 2.) x;;

Most importantly, you can use Owl to iterate a ndarray in various ways. Owl provides a simple but flexible and powerful way to define a "slice" in ndarray. Comparing to the "Bigarray.slice_left" function, the slice in Owl does not have to start from the left-most axis. E.g., for the previously defined [|3;4;5|] ndarray, you can define a slice in the following ways:

let s0 = [ []; []; [] ]      (* (*,*,*), essentially the whole ndarray as one slice *)
let s1 = [ [0]; []; [] ]     (* (0,*,*) *)
let s2 = [ []; [2]; [] ]     (* (*,2,*) *)
let s3 = [ []; []; [1] ]     (* (*,*,1) *)
let s4 = [ [1]; []; [2] ]    (* (1,*,2) *)
...

slice function is very flexible, it basically has the same semantic as that in numpy. So you know how to index ndarray in numpy, you should be able to do the same thing in Owl. For advanced use of slice fucntion, please refer to my separate tutorial. Some examples as as below.

let s = [ [1]; []; [-1;0;-1]; ];;
let s = [ [1]; [0]; [-1;0;-1]; ];;
let s = [ [1]; [0]; [-2;0]; ];;
let s = [ [0]; [0;1]; [-2;0;-2]; ];;
...

With the slice definition above, we can iterate and map the elements in a slice. E.g., we add one to all the elements in slice (0,*,*).

Arr.map ~axis:[ [0]; []; [] ] (fun a -> a +. 1.) x;;

There are more functions to help you to iterate elements and slices in a ndarray: iteri, iter, mapi, map, filteri, filter, foldi, fold, iteri_slice, iter_slice, iter2i, iter2. Please refer to the documentation for their details.

Algorithmic differentiation (AD) is another key component in Owl which can make many analytical tasks so easy to perform. It is also often referred to as Automatic differentiation. Here is a Wikipedia article to help you understand the topic if you are interested in.

The AD support is provided Algodiff module. More precisely, Algodiff.Numerical provides numerical differentiation whilst Algodiff.S and Algodiff.D provides algorithmic differentiation for single and double precision float numbers respectively. For the detailed differences between the two, please read the wiki article as your starting point. Simply put, Algodiff.S/D is able to provide exact result of the derivative whereas Algodiff.Numerical is just approximation which is subject to round and truncate errors.

Algodiff.S supports higher-order derivatives. Here is an example which calculates till the fourth derivative of tanh function.

open Algodiff.S;;

(* calculate derivatives of f0 *)
let f0 x = Maths.(tanh x);;
let f1 = f0 |> diff;;
let f2 = f0 |> diff |> diff;;
let f3 = f0 |> diff |> diff |> diff;;
let f4 = f0 |> diff |> diff |> diff |> diff;;

Quite easy, isn't it? Then we can plot the values of tanh and its four derivatives between interval [-4, 4].

let map f x = Vec.map (fun a -> a |> pack_flt |> f |> unpack_flt) x;;

(* calculate point-wise values *)
let x = Vec.linspace (-4.) 4. 200;;
let y0 = map f0 x;;
let y1 = map f1 x;;
let y2 = map f2 x;;
let y3 = map f3 x;;
let y4 = map f4 x;;

(* plot the values of all functions *)
let h = Plot.create "plot_021.png" in
Plot.set_foreground_color h 0 0 0;
Plot.set_background_color h 255 255 255;
Plot.plot ~h x y0;
Plot.plot ~h x y1;
Plot.plot ~h x y2;
Plot.plot ~h x y3;
Plot.plot ~h x y4;
Plot.output h;;

Then you should be able to see a figure like this one below. For more advanced use, please see my separate tutorial.

Even though this is still work in progress, I find it necessary to present a small neural network example to show how necessary it is to have a comprehensive numerical infrastructure. The illustration in the following is of course the classic MNIST example wherein we will train a two-layer network that can recognise hand-written digits.

Currently Neural module is wrapped into a separate library but it will be merged into Owl main library in the future. First, plese start your utop and load the Owl_neural library.

#require "owl_neural";;
open Owl_neural;;

Now, let's see how to define a two-layer neural network.

let nn = Feedforward.create ();;
Feedforward.add_layer nn (linear 784 300) ~act_typ:Activation.Tanh;;
Feedforward.add_layer nn (linear 300 10) ~act_typ:Activation.Softmax;;

Done! Only three lines of code, that's easy, isn't it? Owl's Neural module is built atop of its Algodiff module. I am often amazed by the power of algorithmic differentiation while developing the neural network module, it just simplifies the design so much and makes life so easy.

Let's look closer at what the code does: the first line defines a Feedforward neural network; the second line adds a linear layer (of shape 784 x 300) with Tanh activation; the third line does the similar thing by adding another linear layer with Softmax activation.

You can print out the summary of the neural network by calling print nn, then you see the following output.

Feedforward network

(0): Linear layer:
  init   : standard
  params : 235500
  w      : 784 x 300
  b      : 1 x 300

(1): Activation layer: tanh

(2): Linear layer:
  init   : standard
  params : 3010
  w      : 300 x 10
  b      : 1 x 10

(3): Activation layer: softmax

How to train the defined network now? You only need two lines of code to load the dataset and start training. By the way, calling Dataset.download_all () will download all the data sets used in Owl (about 1GB uncompressed data).

let x, _, y = Dataset.load_mnist_train_data () in
train nn x y;;

You may ask "what if I want different training configuration?" Well, the training and network module is actually very flexible and highly configurable. But I will talk about these details in another separate tutorial.

If you want to try Owl on ARM based platforms such as Raspberry Pi rather than x86 ones, the installation are similar. Just note that Owl requires OCaml 4.04, which might not be supported on your platform's binary distribution system yet, so you might consider compiling OCaml sources. Besides, to solve a potential conflict with gsl package, after running ./configure in the top directory, you should run:

sed -i -e 's/#define ARCH_ALIGN_DOUBLE/#undef ARCH_ALIGN_DOUBLE/g' config/m.h config/m-templ.h

before running make world.opt.

A Docker image is also provided on Docker Hub specifically for ARM platform. Just pull the image, start a container, then play with it in utop.

docker run --name owl -it matrixanger/owl:arm

Note that after starting a new container you need to run

eval `opam config env`

for once before starting utop.

Owl is under active development, and I really look forward to your comments and contributions. Besides setting up a complete development environment on your native system, the easiest way to contribute is to use the Owl Docker Image. Moreover, we have also built a docker image for ARM-based platform so that you can run Owl on Raspberry PI and Cubietruck (see the section above).

Just pull the image and dig into code saved in /root/owl, then have fun!

Acknowledgement: Funded in part by EPSRC project - Contrive (EP/N028422/1).