Torch (Collobert et al., 2011), Tensorflow (Abadi et al., 2015) and Caffe (Jia
et al., 2014).
This chapter briefly reviews the main building blocks of CNNs, namely dis-
crete convolutions and pooling. For an in-depth treatment of the subject, see
Chapter 9 of the Deep Learning textbook (Goodfellow et al., 2016).
1.1 Discrete convolutions
The bread and butter of neural networks is affine transformations: a vector
is received as input and is multiplied with a matrix to produce an output (to
which a bias vector is usually added before passing the result through a non-
linearity). This is applicable to any type of input, be it an image, a sound
clip or an unordered collection of features: whatever their dimensionality, their
representation can always be flattened into a vector before the transformation.
Images, sound clips and many other similar kinds of data have an intrinsic
structure. More formally, they share these important properties:
• They are stored as multi-dimensional arrays.
• They feature one or more axes for which ordering matters (e.g., width and
height axes for an image, time axis for a sound clip).
• One axis, called the channel axis, is used to access different views of the
data (e.g., the red, green and blue channels of a color image, or the left
and right channels of a stereo audio track).
These properties are not exploited when an affine transformation is applied;
in fact, all the axes are treated in the same way and the topological information
is not taken into account. Still, taking advantage of the implicit structure of
the data may prove very handy in solving some tasks, like computer vision and
speech recognition, and in these cases it would be best to preserve it. This is
where discrete convolutions come into play.
A discrete convolution is a linear transformation that preserves this notion
of ordering. It is sparse (only a few input units contribute to a given output
unit) and reuses parameters (the same weights are applied to multiple locations
in the input).
Figure 1.1 provides an example of a discrete convolution. The light blue
grid is called the input feature map. To keep the drawing simple, a single input
feature map is represented, but it is not uncommon to have multiple feature
maps stacked one onto another.
1
A kernel (shaded area) of value
0 1 2
2 2 0
0 1 2
6
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