.. _sec_ndarray:
Data Manipulation
=================
In order to get anything done, we need some way to store and manipulate
data. Generally, there are two important things we need to do with data:
(i) acquire them; and (ii) process them once they are inside the
computer. There is no point in acquiring data absent some way to store
it, so let’s get our hands dirty first by playing with synthetic data.
To start, we introduce the :math:`n`-dimensional array (``ndarray``),
MXNet’s primary tool for storing and transforming data. In MXNet,
``ndarray`` is a class and we call any instance “an ``ndarray``”.
If you have worked with NumPy, the most widely-used scientific computing
package in Python, then will find this section familiar. That’s by
design. We designed MXNet’s ``ndarray`` to be an extension to NumPy’s
``ndarray`` with a few killer features. First, MXNet’s ``ndarray``
supports asynchronous computation on CPU, GPU, and distributed cloud
architectures, whereas NumPy only supports CPU computation. Second,
MXNet’s ``ndarray`` supports automatic differentiation. These properties
make MXNet’s ``ndarray`` suitable for deep learning. Throughout the
book, when we say ``ndarray``, we are referring to MXNet’s ``ndarray``
unless otherwise stated.
Getting Started
---------------
In this section, we aim to get you up and running, equipping you with
the the basic math and numerical computing tools that you will build on
as you progress through the book. Do not worry if you struggle to grok
some of the mathematical concepts or library functions. The following
sections will revisit this material in the context practical examples
and it will sink. On the other hand, if you already have some background
and want to go deeper into the mathematical content, just skip this
section.
To start, we import the ``np`` (``numpy``) and ``npx``
(``numpy_extension``) modules from MXNet. Here, the ``np`` module
includes functions supported by NumPy, while the ``npx`` module contains
a set of extensions developed to empower deep learning within a
NumPy-like environment. When using ``ndarray``, we almost always invoke
the ``set_np`` function: this is for compatibility of ``ndarray``
processing by other components of MXNet.
.. code:: python
from mxnet import np, npx
npx.set_np()
An ``ndarray`` represents a (possibly multi-dimensional) array of
numerical values. With one axis, an ``ndarray`` corresponds (in math) to
a *vector*. With two axes, an ``ndarray`` corresponds to a *matrix*.
Arrays with more than two axes do not have special mathematical names—we
simply call them *tensors*.
To start, we can use ``arange`` to create a row vector ``x`` containing
the first :math:`12` integers starting with :math:`0`, though they are
created as floats by default. Each of the values in an ``ndarray`` is
called an *element* of the ``ndarray``. For instance, there are
:math:`12` elements in the ``ndarray`` ``x``. Unless otherwise
specified, a new ``ndarray`` will be stored in main memory and
designated for CPU-based computation.
.. code:: python
x = np.arange(12)
x
.. parsed-literal::
:class: output
array([ 0., 1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11.])
We can access an ``ndarray``\ ’s *shape* (the length along each axis) by
inspecting its ``shape`` property.
.. code:: python
x.shape
.. parsed-literal::
:class: output
(12,)
If we just want to know the total number of elements in an ``ndarray``,
i.e., the product of all of the shape elements, we can inspect its
``size`` property. Because we are dealing with a vector here, the single
element of its ``shape`` is identical to its ``size``.
.. code:: python
x.size
.. parsed-literal::
:class: output
12
To change the shape of an ``ndarray`` without altering either the number
of elements or their values, we can invoke the ``reshape`` function. For
example, we can transform our ``ndarray``, ``x``, from a row vector with
shape (:math:`12`,) to a matrix with shape (:math:`3`, :math:`4`). This
new ``ndarray`` contains the exact same values, but views them as a
matrix organized as :math:`3` rows and :math:`4` columns. To reiterate,
although the shape has changed, the elements in ``x`` have not. Note
that the ``size`` is unaltered by reshaping.
.. code:: python
x = x.reshape(3, 4)
x
.. parsed-literal::
:class: output
array([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.]])
Reshaping by manually specifying every dimension is unnecessary. If our
target shape is a matrix with shape (height, width), then after we know
the width, the height is given implicitly. Why should we have to perform
the division ourselves? In the example above, to get a matrix with
:math:`3` rows, we specified both that it should have :math:`3` rows and
:math:`4` columns. Fortunately, ``ndarray`` can automatically work out
one dimension given the rest. We invoke this capability by placing
``-1`` for the dimension that we would like ``ndarray`` to automatically
infer. In our case, instead of calling ``x.reshape(3, 4)``, we could
have equivalently called ``x.reshape(-1, 4)`` or ``x.reshape(3, -1)``.
The ``empty`` method grabs a chunk of memory and hands us back a matrix
without bothering to change the value of any of its entries. This is
remarkably efficient but we must be careful because the entries might
take arbitrary values, including very big ones!
.. code:: python
np.empty((3, 4))
.. parsed-literal::
:class: output
array([[-2.171626e+06, 4.556882e-41, 1.055208e-30, 3.089723e-41],
[ 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00],
[ 0.000000e+00, 0.000000e+00, 0.000000e+00, 0.000000e+00]])
Typically, we will want our matrices initialized either with zeros,
ones, some other constants, or numbers randomly sampled from a specific
distribution. We can create an ``ndarray`` representing a tensor with
all elements set to :math:`0` and a shape of (:math:`2`, :math:`3`,
:math:`4`) as follows:
.. code:: python
np.zeros((2, 3, 4))
.. parsed-literal::
:class: output
array([[[0., 0., 0., 0.],
[0., 0., 0., 0.],
[0., 0., 0., 0.]],
[[0., 0., 0., 0.],
[0., 0., 0., 0.],
[0., 0., 0., 0.]]])
Similarly, we can create tensors with each element set to 1 as follows:
.. code:: python
np.ones((2, 3, 4))
.. parsed-literal::
:class: output
array([[[1., 1., 1., 1.],
[1., 1., 1., 1.],
[1., 1., 1., 1.]],
[[1., 1., 1., 1.],
[1., 1., 1., 1.],
[1., 1., 1., 1.]]])
| Often, we want to randomly sample the values for each element in an
``ndarray`` from some probability distribution. For example, when we
construct arrays to serve as parameters in a neural network, we will
typically inititialize their values randomly. The following snippet
creates an ``ndarray`` with shape (:math:`3`, :math:`4`). Each of its
elements is randomly sampled
| from a standard Gaussian (normal) distribution with a mean of
:math:`0` and a standard deviation of :math:`1`.
.. code:: python
np.random.normal(0, 1, size=(3, 4))
.. parsed-literal::
:class: output
array([[ 2.2122064 , 1.1630787 , 0.7740038 , 0.4838046 ],
[ 1.0434405 , 0.29956347, 1.1839255 , 0.15302546],
[ 1.8917114 , -1.1688148 , -1.2347414 , 1.5580711 ]])
We can also specify the exact values for each element in the desired
``ndarray`` by supplying a Python list (or list of lists) containing the
numerical values. Here, the outermost list corresponds to axis
:math:`0`, and the inner list to axis :math:`1`.
.. code:: python
np.array([[2, 1, 4, 3], [1, 2, 3, 4], [4, 3, 2, 1]])
.. parsed-literal::
:class: output
array([[2., 1., 4., 3.],
[1., 2., 3., 4.],
[4., 3., 2., 1.]])
Operations
----------
This book is not about software engineering. Our interests are not
limited to simply reading and writing data from/to arrays. We want to
perform mathematical operations on those arrays. Some of the simplest
and most useful operations are the *elementwise* operations. These apply
a standard scalar operation to each element of an array. For functions
that take two arrays as inputs, elementwise operations apply some
standard binary operator on each pair of corresponding elements from the
two arrays. We can create an elementwise function from any function that
maps from a scalar to a scalar.
In mathematical notation, we would denote such a *unary* scalar operator
(taking one input) by the signature
:math:`f: \mathbb{R} \rightarrow \mathbb{R}`. This just mean that the
function is mapping from any real number (:math:`\mathbb{R}`) onto
another. Likewise, we denote a *binary* scalar operator (taking two real
inputs, and yielding one output) by the signature
:math:`f: \mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}`. Given any two
vectors :math:`\mathbf{u}` and :math:`\mathbf{v}` *of the same shape*,
and a binary operator :math:`f`, we can produce a vector
:math:`\mathbf{c} = F(\mathbf{u},\mathbf{v})` by setting
:math:`c_i \gets f(u_i, v_i)` for all :math:`i`, where :math:`c_i, u_i`,
and :math:`v_i` are the :math:`i^\mathrm{th}` elements of vectors
:math:`\mathbf{c}, \mathbf{u}`, and :math:`\mathbf{v}`. Here, we
produced the vector-valued
:math:`F: \mathbb{R}^d, \mathbb{R}^d \rightarrow \mathbb{R}^d` by
*lifting* the scalar function to an elementwise vector operation.
In MXNet, the common standard arithmetic operators (``+``, ``-``, ``*``,
``/``, and ``**``) have all been *lifted* to elementwise operations for
any identically-shaped tensors of arbitrary shape. We can call
elementwise operations on any two tensors of the same shape. In the
following example, we use commas to formulate a :math:`5`-element tuple,
where each element is the result of an elementwise operation.
.. code:: python
x = np.array([1, 2, 4, 8])
y = np.array([2, 2, 2, 2])
x + y, x - y, x * y, x / y, x ** y # The ** operator is exponentiation
.. parsed-literal::
:class: output
(array([ 3., 4., 6., 10.]),
array([-1., 0., 2., 6.]),
array([ 2., 4., 8., 16.]),
array([0.5, 1. , 2. , 4. ]),
array([ 1., 4., 16., 64.]))
Many more operations can be applied elementwise, including unary
operators like exponentiation.
.. code:: python
np.exp(x)
.. parsed-literal::
:class: output
array([2.7182817e+00, 7.3890562e+00, 5.4598148e+01, 2.9809580e+03])
In addition to elementwise computations, we can also perform linear
algebra operations, including vector dot products and matrix
multiplication. We will explain the crucial bits of linear algebra (with
no assumed prior knowledge) in :numref:`sec_linear-algebra`.
We can also *concatenate* multiple ``ndarray``\ s together, stacking
them end-to-end to form a larger ``ndarray``. We just need to provide a
list of ``ndarray``\ s and tell the system along which axis to
concatenate. The example below shows what happens when we concatenate
two matrices along rows (axis :math:`0`, the first element of the shape)
vs. columns (axis :math:`1`, the second element of the shape). We can
see that, the first output ``ndarray``\ ‘s axis-\ :math:`0` length
(:math:`6`) is the sum of the two input ``ndarray``\ s’ axis-\ :math:`0`
lengths (:math:`3 + 3`); while the second output ``ndarray``\ ‘s
axis-\ :math:`1` length (:math:`8`) is the sum of the two input
``ndarray``\ s’ axis-\ :math:`1` lengths (:math:`4 + 4`).
.. code:: python
x = np.arange(12).reshape(3, 4)
y = np.array([[2, 1, 4, 3], [1, 2, 3, 4], [4, 3, 2, 1]])
np.concatenate([x, y], axis=0), np.concatenate([x, y], axis=1)
.. parsed-literal::
:class: output
(array([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[ 2., 1., 4., 3.],
[ 1., 2., 3., 4.],
[ 4., 3., 2., 1.]]),
array([[ 0., 1., 2., 3., 2., 1., 4., 3.],
[ 4., 5., 6., 7., 1., 2., 3., 4.],
[ 8., 9., 10., 11., 4., 3., 2., 1.]]))
Sometimes, we want to construct a binary ``ndarray`` via *logical
statements*. Take ``x == y`` as an example. For each position, if ``x``
and ``y`` are equal at that position, the corresponding entry in the new
``ndarray`` takes a value of :math:`1`, meaning that the logical
statement ``x == y`` is true at that position; otherwise that position
takes :math:`0`.
.. code:: python
x == y
.. parsed-literal::
:class: output
array([[False, True, False, True],
[False, False, False, False],
[False, False, False, False]])
Summing all the elements in the ``ndarray`` yields an ``ndarray`` with
only one element.
.. code:: python
x.sum()
.. parsed-literal::
:class: output
array(66.)
For stylistic convenience, we can write ``x.sum()``\ as ``np.sum(x)``.
Broadcasting Mechanism
----------------------
In the above section, we saw how to perform elementwise operations on
two ``ndarray``\ s of the same shape. Under certain conditions, even
when shapes differ, we can still perform elementwise operations by
invoking the *broadcasting mechanism*. These mechanisms work in the
following way: First, expand one or both arrays by copying elements
appropriately so that after this transformation, the two ``ndarray``\ s
have the same shape. Second, carry out the elementwise operations on the
resulting arrays.
In most cases, we broadcast along an axis where an array initially only
has length :math:`1`, such as in the following example:
.. code:: python
a = np.arange(3).reshape(3, 1)
b = np.arange(2).reshape(1, 2)
a, b
.. parsed-literal::
:class: output
(array([[0.],
[1.],
[2.]]), array([[0., 1.]]))
Since ``a`` and ``b`` are :math:`3\times1` and :math:`1\times2` matrices
respectively, their shapes do not match up if we want to add them. We
*broadcast* the entries of both matrices into a larger :math:`3\times2`
matrix as follows: for matrix ``a`` it replicates the columns and for
matrix ``b`` it replicates the rows before adding up both elementwise.
.. code:: python
a + b
.. parsed-literal::
:class: output
array([[0., 1.],
[1., 2.],
[2., 3.]])
Indexing and Slicing
--------------------
Just as in any other Python array, elements in an ``ndarray`` can be
accessed by index. As in any Python array, the first element has index
:math:`0` and ranges are specified to include the first but *before* the
last element. As in standard Python lists, we can access elements
according to their relative position to the end of the list by using
negative indices.
Thus, ``[-1]`` selects the last element and ``[1:3]`` selects the second
and the third elements as follows:
.. code:: python
x[-1], x[1:3]
.. parsed-literal::
:class: output
(array([ 8., 9., 10., 11.]), array([[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.]]))
Beyond reading, we can also write elements of a matrix by specifying
indices.
.. code:: python
x[1, 2] = 9
x
.. parsed-literal::
:class: output
array([[ 0., 1., 2., 3.],
[ 4., 5., 9., 7.],
[ 8., 9., 10., 11.]])
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, where ``:`` takes all the elements
along axis :math:`1` (column). While we discussed indexing for matrices,
this obviously also works for vectors and for tensors of more than
:math:`2` dimensions.
.. code:: python
x[0:2, :] = 12
x
.. parsed-literal::
:class: output
array([[12., 12., 12., 12.],
[12., 12., 12., 12.],
[ 8., 9., 10., 11.]])
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 ``ndarray`` that ``y`` used to point to and
instead point ``y`` 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 will find that ``id(y)`` points to a different
location. That is because Python first evaluates ``y + x``, allocating
new memory for the result and then makes ``y`` point to this new
location in memory.
.. code:: python
before = id(y)
y = y + x
id(y) == before
.. parsed-literal::
:class: output
False
This might be undesirable for two reasons. First, we do not want to run
around allocating memory unnecessarily all the time. In machine
learning, we might have hundreds of megabytes of parameters and update
all of them multiple times per second. Typically, we will want to
perform these updates *in place*. Second, we might point at the same
parameters from multiple variables. If we do not update in place, this
could cause that discarded memory is not released, and make it possible
for parts of our code 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[:] = ``. To illustrate this
concept, we first create a new matrix ``z`` with the same shape as
another ``y``, using ``zeros_like`` to allocate a block of :math:`0`
entries.
.. code:: python
z = np.zeros_like(y)
print('id(z):', id(z))
z[:] = x + y
print('id(z):', id(z))
.. parsed-literal::
:class: output
id(z): 139666709179120
id(z): 139666709179120
If the value of ``x`` is not reused in subsequent computations, we can
also use ``x[:] = x + y`` or ``x += y`` to reduce the memory overhead of
the operation.
.. code:: python
before = id(x)
x += y
id(x) == before
.. parsed-literal::
:class: output
True
Conversion to Other Python Objects
----------------------------------
Converting an MXNet ``ndarray`` to a NumPy ``ndarray``, or vice versa,
is easy. The converted result does not share memory. This minor
inconvenience is actually quite important: when you perform operations
on the CPU or on GPUs, you do not want MXNet to halt computation,
waiting to see whether the NumPy package of Python might want to be
doing something else with the same chunk of memory. The ``array`` and
``asnumpy`` functions do the trick.
.. code:: python
a = x.asnumpy()
b = np.array(a)
type(a), type(b)
.. parsed-literal::
:class: output
(numpy.ndarray, mxnet.numpy.ndarray)
To convert a size-\ :math:`1` ``ndarray`` to a Python scalar, we can
invoke the ``item`` function or Python’s built-in functions.
.. code:: python
a = np.array([3.5])
a, a.item(), float(a), int(a)
.. parsed-literal::
:class: output
(array([3.5]), 3.5, 3.5, 3)
Summary
-------
- MXNet’s ``ndarray`` is an extension to NumPy’s ``ndarray`` with a few
killer advantages that make it suitable for deep learning.
- MXNet’s ``ndarray`` provides a variety of functionalities including
basic mathematics operations, broadcasting, indexing, slicing, memory
saving, and conversion to other Python objects.
Exercises
---------
1. 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.
2. Replace the two ``ndarray``\ s that operate by element in the
broadcasting mechanism with other shapes, e.g., three dimensional
tensors. Is the result the same as expected?
`Discussions `__
-------------------------------------------------
|image0|
.. |image0| image:: ../img/qr_ndarray.svg