.. _chapter_channels:
Multiple Input and Output Channels
==================================
While we have described the multiple channels that comprise each image
(e.g. color images have the standard RGB channels to indicate the amount
of red, green and blue), until now, we simplified all of our numerical
examples by working with just a single input and a single output
channel. This has allowed us to think of our inputs, convolutional
kernels, and outputs each as two-dimensional arrays.
When we add channels into the mix, our inputs and hidden representations
both become three-dimensional arrays. For example, each RGB input image
has shape :math:`3\times h\times w`. We refer to this axis, with a size
of 3, as the channel dimension. In this section, we will take a deeper
look at convolution kernels with multiple input and multiple output
channels.
Multiple Input Channels
-----------------------
When the input data contains multiple channels, we need to construct a
convolution kernel with the same number of input channels as the input
data, so that it can perform cross-correlation with the input data.
Assuming that the number of channels for the input data is :math:`c_i`,
the number of input channels of the convolution kernel also needs to be
:math:`c_i`. If our convolution kernel’s window shape is
:math:`k_h\times k_w`, then when :math:`c_i=1`, we can think of our
convolution kernel as just a two-dimensional array of shape
:math:`k_h\times k_w`.
However, when :math:`c_i>1`, we need a kernel that contains an array of
shape :math:`k_h\times k_w` *for each input channel*. Concatenating
these :math:`c_i` arrays together yields a convolution kernel of shape
:math:`c_i\times k_h\times k_w`. Since the input and convolution kernel
each have :math:`c_i` channels, we can perform a cross-correlation
operation on the two-dimensional array of the input and the
two-dimensional kernel array of the convolution kernel for each channel,
adding the :math:`c_i` results together (summing over the channels) to
yield a two-dimensional array. This is the result of a two-dimensional
cross-correlation between multi-channel input data and a *multi-input
channel* convolution kernel.
In the figure below, we demonstrate an example of a two-dimensional
cross-correlation with two input channels. The shaded portions are the
first output element as well as the input and kernel array elements used
in its computation:
:math:`(1\times1+2\times2+4\times3+5\times4)+(0\times0+1\times1+3\times2+4\times3)=56`.
.. figure:: ../img/conv-multi-in.svg
Cross-correlation computation with 2 input channels. The shaded
portions are the first output element as well as the input and kernel
array elements used in its computation:
:math:`(1\times1+2\times2+4\times3+5\times4)+(0\times0+1\times1+3\times2+4\times3)=56`.
To make sure we reall understand what’s going on here, we can implement
cross-correlation operations with multiple input channels ourselves.
Notice that all we are doing is performing one cross-correlation
operation per channel and then adding up the results using the ``add_n``
function.
.. code:: python
import d2l
from mxnet import nd
def corr2d_multi_in(X, K):
# First, traverse along the 0th dimension (channel dimension) of X and K.
# Then, add them together by using * to turn the result list into a
# positional argument of the add_n function
return nd.add_n(*[d2l.corr2d(x, k) for x, k in zip(X, K)])
We can construct the input array ``X`` and the kernel array ``K``
corresponding to the values in the above diagram to validate the output
of the cross-correlation operation.
.. code:: python
X = nd.array([[[0, 1, 2], [3, 4, 5], [6, 7, 8]],
[[1, 2, 3], [4, 5, 6], [7, 8, 9]]])
K = nd.array([[[0, 1], [2, 3]], [[1, 2], [3, 4]]])
corr2d_multi_in(X, K)
.. parsed-literal::
:class: output
[[ 56. 72.]
[104. 120.]]
Multiple Output Channels
------------------------
Regardless of the number of input channels, so far we always ended up
with one output channel. However, as we discussed earlier, it turns out
to be essential to have multiple channels at each layer. In the most
popular neural network architectures, we actually increase the channel
dimension as we go higher up in the neural network, typically
downsampling to trade off spatial resolution for greater *channel
depth*. Intuitively, you could think of each channel as responding to
some different set of features. Reality is a bit more complicated than
the most naive intepretations of this intuition since representations
aren’t learned independent but are rather optimized to be jointly
useful. So it may not be that a single channel learns an edge detector
but rather that some direction in channel space corresponds to detecting
edges.
Denote by :math:`c_i` and :math:`c_o` the number of input and output
channels, respectively, and let :math:`k_h` and :math:`k_w` be the
height and width of the kernel. To get an output with multiple channels,
we can create a kernel array of shape :math:`c_i\times k_h\times k_w`
for each output channel. We concatenate them on the output channel
dimension, so that the shape of the convolution kernel is
:math:`c_o\times c_i\times k_h\times k_w`. In cross-correlation
operations, the result on each output channel is calculated from the
convolution kernel corrsponding to that output channel and takes input
from all channels in the input array.
We implement a cross-correlation function to calculate the output of
multiple channels as shown below.
.. code:: python
def corr2d_multi_in_out(X, K):
# Traverse along the 0th dimension of K, and each time, perform
# cross-correlation operations with input X. All of the results are merged
# together using the stack function
return nd.stack(*[corr2d_multi_in(X, k) for k in K])
We construct a convolution kernel with 3 output channels by
concatenating the kernel array ``K`` with ``K+1`` (plus one for each
element in ``K``) and ``K+2``.
.. code:: python
K = nd.stack(K, K + 1, K + 2)
K.shape
.. parsed-literal::
:class: output
(3, 2, 2, 2)
Below, we perform cross-correlation operations on the input array ``X``
with the kernel array ``K``. Now the output contains 3 channels. The
result of the first channel is consistent with the result of the
previous input array ``X`` and the multi-input channel, single-output
channel kernel.
.. code:: python
corr2d_multi_in_out(X, K)
.. parsed-literal::
:class: output
[[[ 56. 72.]
[104. 120.]]
[[ 76. 100.]
[148. 172.]]
[[ 96. 128.]
[192. 224.]]]
:math:`1\times 1` Convolutional Layer
-------------------------------------
At first, a :math:`1 \times 1` convolution, i.e. :math:`k_h = k_w = 1`,
doesn’t seem to make much sense. After all, a convolution correlates
adjacent pixels. A :math:`1 \times 1` convolution obviously doesn’t.
Nonetheless, they are popular operations that are sometimes included in
the designs of complex deep networks. Let’s see in some detail what it
actually does.
Because the minimum window is used, the :math:`1\times 1` convolution
loses the ability of larger convolutional layers to recognize patterns
consisting of interactions among adjacent elements in the height and
width dimensions. The only computation of the :math:`1\times 1`
convolution occurs on the channel dimension.
The figure below shows the cross-correlation computation using the
:math:`1\times 1` convolution kernel with 3 input channels and 2 output
channels. Note that the inputs and outputs have the same height and
width. Each element in the output is derived from a linear combination
of elements *at the same position* in the input image. You could think
of the :math:`1\times 1` convolutional layer as constituting a
fully-connected layer applied at every single pixel location to
transform the c_i corresponding input values into c_o output values.
Because this is still a convolutional layer, the weights are tied across
pixel location Thus the :math:`1\times 1` convolutional layer requires
:math:`c_o\times c_i` weights (plus the bias terms).
.. figure:: ../img/conv-1x1.svg
The cross-correlation computation uses the :math:`1\times 1`
convolution kernel with 3 input channels and 2 output channels. The
inputs and outputs have the same height and width.
Let’s check whether this works in practice: we implement the
:math:`1 \times 1` convolution using a fully-connected layer. The only
thing is that we need to make some adjustments to the data shape before
and after the matrix multiplication.
.. code:: python
def corr2d_multi_in_out_1x1(X, K):
c_i, h, w = X.shape
c_o = K.shape[0]
X = X.reshape((c_i, h * w))
K = K.reshape((c_o, c_i))
Y = nd.dot(K, X) # Matrix multiplication in the fully connected layer
return Y.reshape((c_o, h, w))
When performing :math:`1\times 1` convolution, the above function is
equivalent to the previously implemented cross-correlation function
``corr2d_multi_in_out``. Let’s check this with some reference data.
.. code:: python
X = nd.random.uniform(shape=(3, 3, 3))
K = nd.random.uniform(shape=(2, 3, 1, 1))
Y1 = corr2d_multi_in_out_1x1(X, K)
Y2 = corr2d_multi_in_out(X, K)
(Y1 - Y2).norm().asscalar() < 1e-6
.. parsed-literal::
:class: output
True
Summary
-------
- Multiple channels can be used to extend the model parameters of the
convolutional layer.
- The :math:`1\times 1` convolutional layer is equivalent to the
fully-connected layer, when applied on a per pixel basis.
- The :math:`1\times 1` convolutional layer is typically used to adjust
the number of channels between network layers and to control model
complexity.
Exercises
---------
1. Assume that we have two convolutional kernels of size :math:`k_1` and
:math:`k_2` respectively (with no nonlinearity in between).
- Prove that the result of the operation can be expressed by a
single convolution.
- What is the dimensionality of the equivalent single convolution?
- Is the converse true?
2. Assume an input shape of :math:`c_i\times h\times w` and a
convolution kernel with the shape
:math:`c_o\times c_i\times k_h\times k_w`, padding of
:math:`(p_h, p_w)`, and stride of :math:`(s_h, s_w)`.
- What is the computational cost (multiplications and additions) for
the forward computation?
- What is the memory footprint?
- What is the memory footprint for the backward computation?
- What is the computational cost for the backward computation?
3. By what factor does the number of calculations increase if we double
the number of input channels :math:`c_i` and the number of output
channels :math:`c_o`? What happens if we double the padding?
4. If the height and width of the convolution kernel is
:math:`k_h=k_w=1`, what is the complexity of the forward computation?
5. Are the variables ``Y1`` and ``Y2`` in the last example of this
section exactly the same? Why?
6. How would you implement convolutions using matrix multiplication when
the convolution window is not :math:`1\times 1`?
Scan the QR Code to `Discuss `__
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.. |image0| image:: ../img/qr_channels.svg