Densely Connected Networks (DenseNet)
=====================================
ResNet significantly changed the view of how to parametrize the
functions in deep networks. *DenseNet* (dense convolutional network) is
to some extent the logical extension of this
:cite:`Huang.Liu.Van-Der-Maaten.ea.2017`. To understand how to arrive
at it, let us take a small detour to mathematics.
From ResNet to DenseNet
-----------------------
Recall the Taylor expansion for functions. For the point :math:`x = 0`
it can be written as
.. math:: f(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \ldots.
The key point is that it decomposes a function into increasingly higher
order terms. In a similar vein, ResNet decomposes functions into
.. math:: f(\mathbf{x}) = \mathbf{x} + g(\mathbf{x}).
That is, ResNet decomposes :math:`f` into a simple linear term and a
more complex nonlinear one. What if we want to capture (not necessarily
add) information beyond two terms? One solution was DenseNet
:cite:`Huang.Liu.Van-Der-Maaten.ea.2017`.
.. _fig_densenet_block:
.. figure:: ../img/densenet-block.svg
The main difference between ResNet (left) and DenseNet (right) in
cross-layer connections: use of addition and use of concatenation.
As shown in :numref:`fig_densenet_block`, the key difference between
ResNet and DenseNet is that in the latter case outputs are
*concatenated* (denoted by :math:`[,]`) rather than added. As a result,
we perform a mapping from :math:`\mathbf{x}` to its values after
applying an increasingly complex sequence of functions:
.. math::
\mathbf{x} \to \left[
\mathbf{x},
f_1(\mathbf{x}),
f_2([\mathbf{x}, f_1(\mathbf{x})]), f_3([\mathbf{x}, f_1(\mathbf{x}), f_2([\mathbf{x}, f_1(\mathbf{x})])]), \ldots\right].
In the end, all these functions are combined in MLP to reduce the number
of features again. In terms of implementation this is quite simple:
rather than adding terms, we concatenate them. The name DenseNet arises
from the fact that the dependency graph between variables becomes quite
dense. The last layer of such a chain is densely connected to all
previous layers. The dense connections are shown in
:numref:`fig_densenet`.
.. _fig_densenet:
.. figure:: ../img/densenet.svg
Dense connections in DenseNet.
The main components that compose a DenseNet are *dense blocks* and
*transition layers*. The former define how the inputs and outputs are
concatenated, while the latter control the number of channels so that it
is not too large.
Dense Blocks
------------
DenseNet uses the modified "batch normalization, activation, and
convolution" structure of ResNet (see the exercise in
:numref:`sec_resnet`). First, we implement this convolution block
structure.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
from mxnet import np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l
npx.set_np()
def conv_block(num_channels):
blk = nn.Sequential()
blk.add(nn.BatchNorm(),
nn.Activation('relu'),
nn.Conv2D(num_channels, kernel_size=3, padding=1))
return blk
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
import torch
from torch import nn
from d2l import torch as d2l
def conv_block(input_channels, num_channels):
return nn.Sequential(
nn.BatchNorm2d(input_channels), nn.ReLU(),
nn.Conv2d(input_channels, num_channels, kernel_size=3, padding=1))
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
import tensorflow as tf
from d2l import tensorflow as d2l
class ConvBlock(tf.keras.layers.Layer):
def __init__(self, num_channels):
super(ConvBlock, self).__init__()
self.bn = tf.keras.layers.BatchNormalization()
self.relu = tf.keras.layers.ReLU()
self.conv = tf.keras.layers.Conv2D(
filters=num_channels, kernel_size=(3, 3), padding='same')
self.listLayers = [self.bn, self.relu, self.conv]
def call(self, x):
y = x
for layer in self.listLayers.layers:
y = layer(y)
y = tf.keras.layers.concatenate([x,y], axis=-1)
return y
.. raw:: html
.. raw:: html
A *dense block* consists of multiple convolution blocks, each using the
same number of output channels. In the forward propagation, however, we
concatenate the input and output of each convolution block on the
channel dimension.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
class DenseBlock(nn.Block):
def __init__(self, num_convs, num_channels, **kwargs):
super().__init__(**kwargs)
self.net = nn.Sequential()
for _ in range(num_convs):
self.net.add(conv_block(num_channels))
def forward(self, X):
for blk in self.net:
Y = blk(X)
# Concatenate the input and output of each block on the channel
# dimension
X = np.concatenate((X, Y), axis=1)
return X
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
class DenseBlock(nn.Module):
def __init__(self, num_convs, input_channels, num_channels):
super(DenseBlock, self).__init__()
layer = []
for i in range(num_convs):
layer.append(conv_block(
num_channels * i + input_channels, num_channels))
self.net = nn.Sequential(*layer)
def forward(self, X):
for blk in self.net:
Y = blk(X)
# Concatenate the input and output of each block on the channel
# dimension
X = torch.cat((X, Y), dim=1)
return X
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
class DenseBlock(tf.keras.layers.Layer):
def __init__(self, num_convs, num_channels):
super(DenseBlock, self).__init__()
self.listLayers = []
for _ in range(num_convs):
self.listLayers.append(ConvBlock(num_channels))
def call(self, x):
for layer in self.listLayers.layers:
x = layer(x)
return x
.. raw:: html
.. raw:: html
In the following example, we define a ``DenseBlock`` instance with 2
convolution blocks of 10 output channels. When using an input with 3
channels, we will get an output with :math:`3+2\times 10=23` channels.
The number of convolution block channels controls the growth in the
number of output channels relative to the number of input channels. This
is also referred to as the *growth rate*.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = DenseBlock(2, 10)
blk.initialize()
X = np.random.uniform(size=(4, 3, 8, 8))
Y = blk(X)
Y.shape
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
(4, 23, 8, 8)
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = DenseBlock(2, 3, 10)
X = torch.randn(4, 3, 8, 8)
Y = blk(X)
Y.shape
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
torch.Size([4, 23, 8, 8])
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = DenseBlock(2, 10)
X = tf.random.uniform((4, 8, 8, 3))
Y = blk(X)
Y.shape
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
TensorShape([4, 8, 8, 23])
.. raw:: html
.. raw:: html
Transition Layers
-----------------
Since each dense block will increase the number of channels, adding too
many of them will lead to an excessively complex model. A *transition
layer* is used to control the complexity of the model. It reduces the
number of channels by using the :math:`1\times 1` convolutional layer
and halves the height and width of the average pooling layer with a
stride of 2, further reducing the complexity of the model.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
def transition_block(num_channels):
blk = nn.Sequential()
blk.add(nn.BatchNorm(), nn.Activation('relu'),
nn.Conv2D(num_channels, kernel_size=1),
nn.AvgPool2D(pool_size=2, strides=2))
return blk
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
def transition_block(input_channels, num_channels):
return nn.Sequential(
nn.BatchNorm2d(input_channels), nn.ReLU(),
nn.Conv2d(input_channels, num_channels, kernel_size=1),
nn.AvgPool2d(kernel_size=2, stride=2))
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
class TransitionBlock(tf.keras.layers.Layer):
def __init__(self, num_channels, **kwargs):
super(TransitionBlock, self).__init__(**kwargs)
self.batch_norm = tf.keras.layers.BatchNormalization()
self.relu = tf.keras.layers.ReLU()
self.conv = tf.keras.layers.Conv2D(num_channels, kernel_size=1)
self.avg_pool = tf.keras.layers.AvgPool2D(pool_size=2, strides=2)
def call(self, x):
x = self.batch_norm(x)
x = self.relu(x)
x = self.conv(x)
return self.avg_pool(x)
.. raw:: html
.. raw:: html
Apply a transition layer with 10 channels to the output of the dense
block in the previous example. This reduces the number of output
channels to 10, and halves the height and width.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = transition_block(10)
blk.initialize()
blk(Y).shape
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
(4, 10, 4, 4)
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = transition_block(23, 10)
blk(Y).shape
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
torch.Size([4, 10, 4, 4])
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = TransitionBlock(10)
blk(Y).shape
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
TensorShape([4, 4, 4, 10])
.. raw:: html
.. raw:: html
DenseNet Model
--------------
Next, we will construct a DenseNet model. DenseNet first uses the same
single convolutional layer and maximum pooling layer as in ResNet.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
net = nn.Sequential()
net.add(nn.Conv2D(64, kernel_size=7, strides=2, padding=3),
nn.BatchNorm(), nn.Activation('relu'),
nn.MaxPool2D(pool_size=3, strides=2, padding=1))
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
b1 = nn.Sequential(
nn.Conv2d(1, 64, kernel_size=7, stride=2, padding=3),
nn.BatchNorm2d(64), nn.ReLU(),
nn.MaxPool2d(kernel_size=3, stride=2, padding=1))
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
def block_1():
return tf.keras.Sequential([
tf.keras.layers.Conv2D(64, kernel_size=7, strides=2, padding='same'),
tf.keras.layers.BatchNormalization(),
tf.keras.layers.ReLU(),
tf.keras.layers.MaxPool2D(pool_size=3, strides=2, padding='same')])
.. raw:: html
.. raw:: html
Then, similar to the four modules made up of residual blocks that ResNet
uses, DenseNet uses four dense blocks. Similar to ResNet, we can set the
number of convolutional layers used in each dense block. Here, we set it
to 4, consistent with the ResNet-18 model in :numref:`sec_resnet`.
Furthermore, we set the number of channels (i.e., growth rate) for the
convolutional layers in the dense block to 32, so 128 channels will be
added to each dense block.
In ResNet, the height and width are reduced between each module by a
residual block with a stride of 2. Here, we use the transition layer to
halve the height and width and halve the number of channels.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
# `num_channels`: the current number of channels
num_channels, growth_rate = 64, 32
num_convs_in_dense_blocks = [4, 4, 4, 4]
for i, num_convs in enumerate(num_convs_in_dense_blocks):
net.add(DenseBlock(num_convs, growth_rate))
# This is the number of output channels in the previous dense block
num_channels += num_convs * growth_rate
# A transition layer that halves the number of channels is added between
# the dense blocks
if i != len(num_convs_in_dense_blocks) - 1:
num_channels //= 2
net.add(transition_block(num_channels))
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
# `num_channels`: the current number of channels
num_channels, growth_rate = 64, 32
num_convs_in_dense_blocks = [4, 4, 4, 4]
blks = []
for i, num_convs in enumerate(num_convs_in_dense_blocks):
blks.append(DenseBlock(num_convs, num_channels, growth_rate))
# This is the number of output channels in the previous dense block
num_channels += num_convs * growth_rate
# A transition layer that halves the number of channels is added between
# the dense blocks
if i != len(num_convs_in_dense_blocks) - 1:
blks.append(transition_block(num_channels, num_channels // 2))
num_channels = num_channels // 2
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
def block_2():
net = block_1()
# `num_channels`: the current number of channels
num_channels, growth_rate = 64, 32
num_convs_in_dense_blocks = [4, 4, 4, 4]
for i, num_convs in enumerate(num_convs_in_dense_blocks):
net.add(DenseBlock(num_convs, growth_rate))
# This is the number of output channels in the previous dense block
num_channels += num_convs * growth_rate
# A transition layer that halves the number of channels is added
# between the dense blocks
if i != len(num_convs_in_dense_blocks) - 1:
num_channels //= 2
net.add(TransitionBlock(num_channels))
return net
.. raw:: html
.. raw:: html
Similar to ResNet, a global pooling layer and a fully-connected layer
are connected at the end to produce the output.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
net.add(nn.BatchNorm(),
nn.Activation('relu'),
nn.GlobalAvgPool2D(),
nn.Dense(10))
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
net = nn.Sequential(
b1, *blks,
nn.BatchNorm2d(num_channels), nn.ReLU(),
nn.AdaptiveAvgPool2d((1, 1)),
nn.Flatten(),
nn.Linear(num_channels, 10))
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
def net():
net = block_2()
net.add(tf.keras.layers.BatchNormalization())
net.add(tf.keras.layers.ReLU())
net.add(tf.keras.layers.GlobalAvgPool2D())
net.add(tf.keras.layers.Flatten())
net.add(tf.keras.layers.Dense(10))
return net
.. raw:: html
.. raw:: html
Training
--------
Since we are using a deeper network here, in this section, we will
reduce the input height and width from 224 to 96 to simplify the
computation.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr, num_epochs, batch_size = 0.1, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
loss 0.142, train acc 0.947, test acc 0.898
5383.6 examples/sec on gpu(0)
.. figure:: output_densenet_e82156_99_1.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr, num_epochs, batch_size = 0.1, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
loss 0.142, train acc 0.948, test acc 0.882
5574.2 examples/sec on cuda:0
.. figure:: output_densenet_e82156_102_1.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr, num_epochs, batch_size = 0.1, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
loss 0.137, train acc 0.951, test acc 0.895
5709.2 examples/sec on /GPU:0
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
.. figure:: output_densenet_e82156_105_2.svg
.. raw:: html
.. raw:: html
Summary
-------
- In terms of cross-layer connections, unlike ResNet, where inputs and
outputs are added together, DenseNet concatenates inputs and outputs
on the channel dimension.
- The main components that compose DenseNet are dense blocks and
transition layers.
- We need to keep the dimensionality under control when composing the
network by adding transition layers that shrink the number of
channels again.
Exercises
---------
1. Why do we use average pooling rather than maximum pooling in the
transition layer?
2. One of the advantages mentioned in the DenseNet paper is that its
model parameters are smaller than those of ResNet. Why is this the
case?
3. One problem for which DenseNet has been criticized is its high memory
consumption.
1. Is this really the case? Try to change the input shape to
:math:`224\times 224` to see the actual GPU memory consumption.
2. Can you think of an alternative means of reducing the memory
consumption? How would you need to change the framework?
4. Implement the various DenseNet versions presented in Table 1 of the
DenseNet paper :cite:`Huang.Liu.Van-Der-Maaten.ea.2017`.
5. Design an MLP-based model by applying the DenseNet idea. Apply it to
the housing price prediction task in :numref:`sec_kaggle_house`.
.. raw:: html
.. raw:: html
`Discussions `__
.. raw:: html
.. raw:: html
`Discussions `__
.. raw:: html
.. raw:: html
`Discussions `__
.. raw:: html
.. raw:: html