Manipulating Data with `ndarray`¶

It’s impossible to get anything done if we can’t manipulate data. Generally, there are two important things we need to do with: (i) acquire it and (ii) process it once it’s inside the computer. There’s no point in trying to acquire data if we don’t even know how to store it, so let’s get our hands dirty first by playing with synthetic data.

We’ll start by introducing NDArrays, MXNet’s primary tool for storing and transforming data. If you’ve worked with NumPy before, you’ll notice that NDArrays are, by design, similar to NumPy’s multi-dimensional array. However, they confer a few key advantages. First, NDArrays support asynchronous computation on CPU, GPU, and distributed cloud architectures. Second, they provide support for automatic differentiation. These properties make NDArray an ideal ingredient for machine learning.

Getting Started¶

In this chapter, we’ll get you going with the basic functionality. Don’t worry if you don’t understand any of the basic math, like element-wise operations or normal distributions. In the next two chapters we’ll take another pass at NDArray, teaching you both the math you’ll need and how to realize it in code. For even more math, see the “Math” section in the appendix.

We begin by importing MXNet and the ndarray module from MXNet. Here, nd is short for ndarray.

In [1]:

import mxnet as mx
from mxnet import nd

The simplest object we can create is a vector. arange creates a row vector of 12 consecutive integers.

In [2]:

x = nd.arange(12)
x

Out[2]:

[ 0.  1.  2.  3.  4.  5.  6.  7.  8.  9. 10. 11.]
<NDArray 12 @cpu(0)>

From the property <NDArray 12 @cpu(0)> shown when printing x we can see that it is a one-dimensional array of length 12 and that it resides in CPU main memory. The 0 in `@cpu(0)`` has no special meaning and does not represent a specific core.

We can get the NDArray instance shape through the shape property.

In [3]:

x.shape

Out[3]:

(12,)

We can also get the total number of elements in the NDArray instance through the size property. Since we are dealing with a vector, both are identical.

In [4]:

x.size

Out[4]:

In the following, we use the reshape function to change the shape of the line vector x to (3, 4), which is a matrix of 3 rows and 4 columns. Except for the shape change, the elements in x (and also its size) remain unchanged.

In [5]:

x = x.reshape((3, 4))
x

Out[5]:

[[ 0.  1.  2.  3.]
 [ 4.  5.  6.  7.]
 [ 8.  9. 10. 11.]]
<NDArray 3x4 @cpu(0)>

It can be awkward to reshape a matrix in the way described above. After all, if we want a matrix with 3 rows we also need to know that it should have 4 columns in order to make up 12 elements. Or we might want to request NDArray to figure out automatically how to give us a matrix with 4 columns and whatever number of rows that are needed to take care of all elements. This is precisely what the entry -1 does in any one of the fields. That is, in our case x.reshape((3, 4)) is equivalent to x.reshape((-1, 4)) and x.reshape((3, -1)).

In [6]:

nd.empty((3, 4))

Out[6]:

[[-5.0071655e+02  4.5678126e-41  1.2883097e-26  3.0769712e-41]
 [ 0.0000000e+00  0.0000000e+00  0.0000000e+00  0.0000000e+00]
 [ 0.0000000e+00  0.0000000e+00  0.0000000e+00  0.0000000e+00]]
<NDArray 3x4 @cpu(0)>

The empty method just grabs some memory and hands us back a matrix without setting the values of any of its entries. This is very efficient but it means that the entries can have any form of values, including very big ones! But typically, we’ll want our matrices initialized.

Commonly, we want one of all zeros. For objects with more than two dimensions mathematicians don’t have special names - they simply call them tensors. To create one with all elements set to 0 a shape of (2, 3, 4) we use

In [7]:

nd.zeros((2, 3, 4))

Out[7]:

[[[0. 0. 0. 0.]
  [0. 0. 0. 0.]
  [0. 0. 0. 0.]]

 [[0. 0. 0. 0.]
  [0. 0. 0. 0.]
  [0. 0. 0. 0.]]]
<NDArray 2x3x4 @cpu(0)>

Just like in numpy, creating tensors with each element being 1 works via

In [8]:

nd.ones((2, 3, 4))

Out[8]:

[[[1. 1. 1. 1.]
  [1. 1. 1. 1.]
  [1. 1. 1. 1.]]

 [[1. 1. 1. 1.]
  [1. 1. 1. 1.]
  [1. 1. 1. 1.]]]
<NDArray 2x3x4 @cpu(0)>

We can also specify the value of each element in the NDArray that needs to be created through a Python list.

In [9]:

y = nd.array([[2, 1, 4, 3], [1, 2, 3, 4], [4, 3, 2, 1]])
y

Out[9]:

[[2. 1. 4. 3.]
 [1. 2. 3. 4.]
 [4. 3. 2. 1.]]
<NDArray 3x4 @cpu(0)>

In some cases, we need to randomly generate the value of each element in the NDArray. This is especially common when we intend to use the array as a parameter in a neural network. The following creates an NDArray with a shape of (3,4). Each of its elements is randomly sampled in a normal distribution with zero mean and unit variance.

In [10]:

nd.random.normal(0, 1, shape=(3, 4))

Out[10]:

[[ 2.2122064   0.7740038   1.0434405   1.1839255 ]
 [ 1.8917114  -1.2347414  -1.771029   -0.45138445]
 [ 0.57938355 -1.856082   -1.9768796  -0.20801921]]
<NDArray 3x4 @cpu(0)>

Operations¶

Oftentimes, we want to apply functions to arrays. Some of the simplest and most useful functions are the element-wise functions. These operate by performing a single scalar operation on the corresponding elements of two arrays. We can create an element-wise function from any function that maps from the scalars to the scalars. In math notations we would denote such a function as \(f: \mathbb{R} \rightarrow \mathbb{R}\). Given any two vectors \(\mathbf{u}\) and \(\mathbf{v}\) of the same shape, and the function f, we can produce a vector \(\mathbf{c} = F(\mathbf{u},\mathbf{v})\) by setting \(c_i \gets f(u_i, v_i)\) for all \(i\). Here, we produced the vector-valued \(F: \mathbb{R}^d \rightarrow \mathbb{R}^d\) by lifting the scalar function to an element-wise vector operation. In MXNet, the common standard arithmetic operators (+,-,/,*,**) have all been lifted to element-wise operations for identically-shaped tensors of arbitrary shape. We can call element-wise operations on any two tensors of the same shape, including matrices.

In [11]:

x = nd.array([1, 2, 4, 8])
y = nd.ones_like(x) * 2
print('x =', x)
print('x + y', x + x)
print('x - y', x - x)
print('x * y', x * x)
print('x / y', x / x)

x =
[1. 2. 4. 8.]
<NDArray 4 @cpu(0)>
x + y
[ 2.  4.  8. 16.]
<NDArray 4 @cpu(0)>
x - y
[0. 0. 0. 0.]
<NDArray 4 @cpu(0)>
x * y
[ 1.  4. 16. 64.]
<NDArray 4 @cpu(0)>
x / y
[1. 1. 1. 1.]
<NDArray 4 @cpu(0)>

Many more operations can be applied element-wise, such as exponentiation:

In [12]:

x.exp()

Out[12]:

[2.7182817e+00 7.3890562e+00 5.4598148e+01 2.9809580e+03]
<NDArray 4 @cpu(0)>

In addition to computations by element, we can also use the dot function for matrix operations. Next, we will perform matrix multiplication to transpose x and y. We define x as a matrix of 3 rows and 4 columns, and y is transposed into a matrix of 4 rows and 3 columns. The two matrices are multiplied to obtain a matrix of 3 rows and 3 columns (if you’re confused about what this means, don’t worry - we will explain matrix operations in much more detail in the chapter on linear algebra).

In [13]:

x = nd.arange(12).reshape((3,4))
y = nd.array([[2, 1, 4, 3], [1, 2, 3, 4], [4, 3, 2, 1]])
nd.dot(x, y.T)

Out[13]:

[[ 18.  20.  10.]
 [ 58.  60.  50.]
 [ 98. 100.  90.]]
<NDArray 3x3 @cpu(0)>

We can also merge multiple NDArrays. For that we need to tell the system along with dimension to merge. The example below merges two matrices along dimension 0 (along rows) and dimension 1 (along columns) respectively.

In [14]:

nd.concat(x, y, dim=0)
nd.concat(x, y, dim=1)

Out[14]:

[[ 0.  1.  2.  3.  2.  1.  4.  3.]
 [ 4.  5.  6.  7.  1.  2.  3.  4.]
 [ 8.  9. 10. 11.  4.  3.  2.  1.]]
<NDArray 3x8 @cpu(0)>

Just like in Numpy, we can construct binary NDarrays by a logical statement. Take x == y as an example. If x and y are equal for some entry, the new NDArray has a value of 1 at the same position; otherwise, it is 0.

In [15]:

x == y

Out[15]:

[[0. 1. 0. 1.]
 [0. 0. 0. 0.]
 [0. 0. 0. 0.]]
<NDArray 3x4 @cpu(0)>

Summing all the elements in the NDArray yields an NDArray with only one element.

In [16]:

x.sum()

Out[16]:

[66.]
<NDArray 1 @cpu(0)>

We can transform the result into a scalar in Python using the asscalar function. In the following example, the \(\ell_2\) norm of x yields a single element NDArray. The final result is transformed into a scalar.

In [17]:

x.norm().asscalar()

Out[17]:

22.494444

For stylistic convenience, we can write y.exp(), x.sum(), x.norm(), etc. also as nd.exp(y), nd.sum(x), nd.norm(x).

Broadcast Mechanism¶

In the above section, we saw how to perform operations on two NDArrays of the same shape. When their shapes differ, a broadcasting mechanism may be triggered analogous to NumPy: first, copy the elements appropriately so that the two NDArrays have the same shape, and then carry out operations by element.

In [18]:

a = nd.arange(3).reshape((3, 1))
b = nd.arange(2).reshape((1, 2))
a, b

Out[18]:

(
 [[0.]
  [1.]
  [2.]]
 <NDArray 3x1 @cpu(0)>,
 [[0. 1.]]
 <NDArray 1x2 @cpu(0)>)

Since a and b are (3x1) and (1x2) matrices respectively, their shapes do not match up if we want to add them. NDArray addresses this by ‘broadcasting’ the entries of both matrices into a larger (3x2) matrix as follows: for matrix a it replicates the columns, for matrix b it replicates the rows before adding up both element-wise.

In [19]:

a + b

Out[19]:

[[0. 1.]
 [1. 2.]
 [2. 3.]]
<NDArray 3x2 @cpu(0)>

Indexing and Slicing¶

Just like in any other Python array, elements in an NDArray can be accessed by its index. In good Python tradition the first element has index 0 and ranges are specified to include the first but not the last. By this logic 1:3 selects the second and third element. Let’s try this out by selecting the respective rows in a matrix.

In [20]:

x[1:3]

Out[20]:

[[ 4.  5.  6.  7.]
 [ 8.  9. 10. 11.]]
<NDArray 2x4 @cpu(0)>

Beyond reading we can also write elements of a matrix.

In [21]:

x[1, 2] = 9
x

Out[21]:

[[ 0.  1.  2.  3.]
 [ 4.  5.  9.  7.]
 [ 8.  9. 10. 11.]]
<NDArray 3x4 @cpu(0)>

If we want to assign multiple elements the same value, we simply index all of them and then assign them the value. For instance, [0:2, :] accesses the first and second rows. While we discussed indexing for matrices, this obviously also works for vectors and for tensors of more than 2 dimensions.

In [22]:

x[0:2, :] = 12
x

Out[22]:

[[12. 12. 12. 12.]
 [12. 12. 12. 12.]
 [ 8.  9. 10. 11.]]
<NDArray 3x4 @cpu(0)>

Saving Memory¶

In the previous example, every time we ran an operation, we allocated new memory to host its results. For example, if we write y = x + y, we will dereference the matrix that y used to point to and instead point it at the newly allocated memory. In the following example we demonstrate this with Python’s id() function, which gives us the exact address of the referenced object in memory. After running y = y + x, we’ll find that id(y) points to a different location. That’s because Python first evaluates y + x, allocating new memory for the result and then subsequently redirects y to point at this new location in memory.

In [23]:

before = id(y)
y = y + x
id(y) == before

Out[23]:

False

This might be undesirable for two reasons. First, we don’t want to run around allocating memory unnecessarily all the time. In machine learning, we might have hundreds of megabytes of paramaters and update all of them multiple times per second. Typically, we’ll want to perform these updates in place. Second, we might point at the same parameters from multiple variables. If we don’t update in place, this could cause a memory leak, and could cause us to inadvertently reference stale parameters.

Fortunately, performing in-place operations in MXNet is easy. We can assign the result of an operation to a previously allocated array with slice notation, e.g., y[:] = <expression>. To illustrate the behavior, we first clone the shape of a matrix using zeros_like to allocate a block of 0 entries.

In [24]:

z = y.zeros_like()
print('id(z):', id(z))
z[:] = x + y
print('id(z):', id(z))

id(z): 140005706155792
id(z): 140005706155792

While this looks pretty, x+y here will still allocate a temporary buffer to store the result of x+y before copying it to y[:]. To make even better use of memory, we can directly invoke the underlying ndarray operation, in this case elemwise_add, avoiding temporary buffers. We do this by specifying the out keyword argument, which every ndarray operator supports:

In [25]:

before = id(z)
nd.elemwise_add(x, y, out=z)
id(z) == before

Out[25]:

True

If the value of x is not reused in subsequent programs, we can also use x[:] = x + y or x += y to reduce the memory overhead of the operation.

In [26]:

before = id(x)
x += y
id(x) == before

Out[26]:

True

Mutual Transformation of NDArray and NumPy¶

Converting MXNet NDArrays to and from NumPy is easy. The converted arrays do not share memory. This minor inconvenience is actually quite important: when you perform operations on the CPU or one of the GPUs, you don’t want MXNet having to wait whether NumPy might want to be doing something else with the same chunk of memory. The array and asnumpy functions do the trick.

In [27]:

import numpy as np

a = x.asnumpy()
print(type(a))
b = nd.array(a)
print(type(b))

<class 'numpy.ndarray'>
<class 'mxnet.ndarray.ndarray.NDArray'>

Problems¶

Run the code in this section. Change the conditional statement x == y in this section to x < y or x > y, and then see what kind of NDArray you can get.
Replace the two NDArrays that operate by element in the broadcast mechanism with other shapes, e.g. three dimensional tensors. Is the result the same as expected?
Assume that we have three matrices a, b and c. Rewrite c = nd.dot(a, b.T) + c in the most memory efficient manner.

Manipulating Data with ndarray¶