Convolutions are an important tool in modern deep neural networks (DNNs). This

post is going to discuss some common types of convolutions, specifically

regular and depthwise separable convolutions. My focus will be on the

implementation of these operation, showing from-scratch Numpy-based code to

compute them and diagrams that explain how things work.

Note that my main goal here is to explain how depthwise separable convolutions

differ from regular ones; if you’re completely new to convolutions I suggest

reading some more introductory resources first.

The code here is compatible with TensorFlow’s definition of convolutions in

the tf.nn module. After

reading this post, the documentation of TensorFlow’s convolution ops should be

easy to decipher.

## Basic 2D convolution

The basic idea behind a 2D convolution is sliding a small window (usually called

a “filter”) over a larger 2D array, and performing a dot product between the

filter elements and the corresponding input array elements at every position.

Here’s a diagram demonstrating the application of a 3×3 convolution filter to

a 6×6 array, in 3 different positions. `W` is the filter, and the yellow-ish

array on the right is the result; the red square shows which element in the

result array is being computed.

Single-channel 2D convolution

The topmost diagram shows the important concept of *padding*: what should we do

when the window goes “out of bounds” on the input array. There are several

options, with the following two being most common in DNNs:

*Valid*padding: in which only valid, in-bounds windows are considered. This

also makes the output smaller than the input, because border elements can’t be

in the center of a filter (unless the filter is 1×1).*Same*padding: in which we assume there’s some constant value outside the

bounds of the input (usually 0) and the filter is applied to every element.

In this case the output array has the same size as the input array. The

diagrams above depict same padding, which I’ll keep using throughout the post.

There are other options for the basic 2D convolution case. For example, the

filter can be moving over the input in jumps of more than 1, thus not centering

on all elements. This is called *stride*, and in this post I’m always using

stride of 1. Convolutions can also be dilated (or *atrous*), wherein the

filter is expanded with gaps between every element. In this post I’m not going

to discuss dilated convolutions and other options – there are plenty of

resources on these topics online.

## Implementing the 2D convolution

Here is a full Python implementation of the simple 2D convolution. It’s called

“single channel” to distinguish it from the more general case in which the input

has more than two dimensions; we’ll get to that shortly.

This implementation is fully self-contained, and only needs Numpy to work. All

the loops are fully explicit – I specifically avoided vectorizing them for

efficiency to maintain clarity:

```
def conv2d_single_channel(input, w):
"""Two-dimensional convolution of a single channel.
Uses SAME padding with 0s, a stride of 1 and no dilation.
input: input array with shape (height, width)
w: filter array with shape (fd, fd) with odd fd.
Returns a result with the same shape as input.
"""
assert w.shape[0] == w.shape[1] and w.shape[0] % 2 == 1
# SAME padding with zeros: creating a new padded array to simplify index
# calculations and to avoid checking boundary conditions in the inner loop.
# padded_input is like input, but padded on all sides with
# half-the-filter-width of zeros.
padded_input = np.pad(input,
pad_width=w.shape[0] // 2,
mode='constant',
constant_values=0)
output = np.zeros_like(input)
for i in range(output.shape[0]):
for j in range(output.shape[1]):
# This inner double loop computes every output element, by
# multiplying the corresponding window into the input with the
# filter.
for fi in range(w.shape[0]):
for fj in range(w.shape[1]):
output[i, j] += padded_input[i + fi, j + fj] * w[fi, fj]
return output
```

## Convolutions in 3 and 4 dimensions

The convolution computed above works in two dimensions; yet, most convolutions

used in DNNs are 4-dimensional. For example, TensorFlow’s `tf.nn.conv2d` op

takes a 4D input tensor and a 4D filter tensor. How come?

The two additional dimensions in the input tensor are *channel* and *batch*. A

canonical example of channels is color images in RGB format. Each pixel has a

value for red, green and blue – three channels overall. So instead of seeing it

as a matrix of triples, we can see it as a 3D tensor where one dimension is

height, another width and another channel (also called the *depth* dimension).

Batch is somewhat different. ML training – with stochastic gradient descent –

is often done in batches for performance; we train the model not on a single

sample at a time, but a “batch” of samples, usually some power of two.

Performing all the operations in tandem on a batch of data makes it easier to

leverage the SIMD capabilities of modern processors. So it doesn’t have any

mathematical significance here – it can be seen as an outer loop over all

operations, performing them for a set of inputs and producing a corresponding

set of outputs.

For filters, the 4 dimensions are height, width, input channel and output

channel. Input channel is the same as the input tensor’s; output channel

collects multiple filters, each of which can be different.

This can be slightly difficult to grasp from text, so here’s a diagram:

Multi-channel 2D convolution

In the diagram and the implementation I’m going to ignore the batch dimension,

since it’s not really mathematically interesting. So the input image has three

dimensions – in this diagram height and width are 8 and depth is 3. The filter

is 3×3 with depth 3. In each step, the filter is slid over the input *in two
dimensions*, and all of its elements are multiplied with the corresponding

elements in the input. That’s 3x3x3=27 multiplications added into the output

element.

Note that this is different from a 3D convolution, where a filter is moved

across the input in all 3 dimensions; true 3D convolutions are not widely used

in DNNs at this time.

So, to reitarate, to compute the multi-channel convolution as shown in the

diagram above, we compute each of the 64 output elements by a dot-product of the

filter with the relevant parts of the input tensor. This produces a single

output channel. To produce additional output channels, we perform the

convolution with additional filters. So if our filter has dimensions (3, 3, 3,

4) this means 4 different 3x3x3 filters. The output will thus have dimensions

8×8 for the spatials and 4 for depth.

Here’s the Numpy implementation of this algorithm:

```
def conv2d_multi_channel(input, w):
"""Two-dimensional convolution with multiple channels.
Uses SAME padding with 0s, a stride of 1 and no dilation.
input: input array with shape (height, width, in_depth)
w: filter array with shape (fd, fd, in_depth, out_depth) with odd fd.
in_depth is the number of input channels, and has the be the same as
input's in_depth; out_depth is the number of output channels.
Returns a result with shape (height, width, out_depth).
"""
assert w.shape[0] == w.shape[1] and w.shape[0] % 2 == 1
padw = w.shape[0] // 2
padded_input = np.pad(input,
pad_width=((padw, padw), (padw, padw), (0, 0)),
mode='constant',
constant_values=0)
height, width, in_depth = input.shape
assert in_depth == w.shape[2]
out_depth = w.shape[3]
output = np.zeros((height, width, out_depth))
for out_c in range(out_depth):
# For each output channel, perform 2d convolution summed across all
# input channels.
for i in range(height):
for j in range(width):
# Now the inner loop also works across all input channels.
for c in range(in_depth):
for fi in range(w.shape[0]):
for fj in range(w.shape[1]):
w_element = w[fi, fj, c, out_c]
output[i, j, out_c] += (
padded_input[i + fi, j + fj, c] * w_element)
return output
```

An interesting point to note here w.r.t. TensorFlow’s `tf.nn.conv2d` op. If

you read its semantics you’ll see discussion of *layout* or *data format*, which

is `NHWC` by default. NHWC simply means the order of dimensions in a 4D

tensor is:

**N**: batch**H**: height (spatial dimension)**W**: width (spatial dimension)**C**: channel (depth)

`NHWC` is the default layout for TensorFlow; another commonly used layout is

`NCHW`, because it’s the format preferred by NVIDIA’s DNN libraries. The code

samples here follow the default.

## Depthwise convolution

Depthwise convolutions are a variation on the operation discussed so far. In the

regular 2D convolution performed over multiple input channels, the filter is as

deep as the input and lets us freely mix channels to generate each element in

the output. Depthwise convolutions don’t do that – each channel is kept separate

– hence the name *depthwise*. Here’s a diagram to help explain how that works:

Depthwise 2D convolution

There are three conceptual stages here:

- Split the input into channels, and split the filter into channels (the number

of channels between input and filter must match). - For each of the channels, convolve the input with the corresponding filter,

producing an output tensor (2D). - Stack the output tensors back together.

Here’s the code implementing it:

```
def depthwise_conv2d(input, w):
"""Two-dimensional depthwise convolution.
Uses SAME padding with 0s, a stride of 1 and no dilation. A single output
channel is used per input channel (channel_multiplier=1).
input: input array with shape (height, width, in_depth)
w: filter array with shape (fd, fd, in_depth)
Returns a result with shape (height, width, in_depth).
"""
assert w.shape[0] == w.shape[1] and w.shape[0] % 2 == 1
padw = w.shape[0] // 2
padded_input = np.pad(input,
pad_width=((padw, padw), (padw, padw), (0, 0)),
mode='constant',
constant_values=0)
height, width, in_depth = input.shape
assert in_depth == w.shape[2]
output = np.zeros((height, width, in_depth))
for c in range(in_depth):
# For each input channel separately, apply its corresponsing filter
# to the input.
for i in range(height):
for j in range(width):
for fi in range(w.shape[0]):
for fj in range(w.shape[1]):
w_element = w[fi, fj, c]
output[i, j, c] += (
padded_input[i + fi, j + fj, c] * w_element)
return output
```

In TensorFlow, the corresponding op is `tf.nn.depthwise_conv2d`; this op has

the notion of *channel multiplier* which lets us compute multiple outputs for

each input channel (somewhat like the number of output channels concept in

`conv2d`).

## Depthwise separable convolution

The depthwise convolution shown above is more commonly used in combination with

an additional step to mix in the channels – *depthwise separable convolution*

[1]:

Depthwise separable convolution

After completing the depthwise convolution, and additional step is performed: a

1×1 convolution across channels. This is exactly the same operation as the

“convolution in 3 dimensions discussed earlier” – just with a 1×1 spatial

filter. This step can be repeated multiple times for different output channels.

The output channels all take the output of the depthwise step and mix it up

with different 1×1 convolutions. Here’s the implementation:

```
def separable_conv2d(input, w_depth, w_pointwise):
"""Depthwise separable convolution.
Performs 2d depthwise convolution with w_depth, and then applies a pointwise
1x1 convolution with w_pointwise on the result.
Uses SAME padding with 0s, a stride of 1 and no dilation. A single output
channel is used per input channel (channel_multiplier=1) in w_depth.
input: input array with shape (height, width, in_depth)
w_depth: depthwise filter array with shape (fd, fd, in_depth)
w_pointwise: pointwise filter array with shape (in_depth, out_depth)
Returns a result with shape (height, width, out_depth).
"""
# First run the depthwise convolution. Its result has the same shape as
# input.
depthwise_result = depthwise_conv2d(input, w_depth)
height, width, in_depth = depthwise_result.shape
assert in_depth == w_pointwise.shape[0]
out_depth = w_pointwise.shape[1]
output = np.zeros((height, width, out_depth))
for out_c in range(out_depth):
for i in range(height):
for j in range(width):
for c in range(in_depth):
w_element = w_pointwise[c, out_c]
output[i, j, out_c] += depthwise_result[i, j, c] * w_element
return output
```

In TensorFlow, this op is called `tf.nn.separable_conv2d`. Similarly to our

implementation it takes two different filter parameters: `depthwise_filter`

for the depthwise step and `pointwise_filter` for the mixing step.

Depthwise separable convolutions have become popular in DNN models recently, for

two reasons:

- They have fewer parameters than “regular” convolutional layers, and thus are

less prone to overfitting. - With fewer parameters, they also require less operations to compute, and thus

are cheaper and faster.

Let’s examine the difference between the number of parameters first. We’ll start

with some definitions:

`S`: spatial dimension – width and height, assuming square inputs.`F`: filter width and height, assuming square filter.`inC`: number of input channels.`outC`: number of output channels.

We also assume `SAME` padding as discussed above, so that the spatial size

of the output matches the input.

In a regular convolution there are `F*F*inC*outC` parameters, because every

filter is 3D and there’s one such filter per output channel.

In depthwise separable convolutions there are `F*F*inC` parameters for the

depthwise part, and then `inC*outC` parameters for the mixing part. It should

be obvious that for a non-trivial `outC`, the sum of these two is significanly

smaller than `F*F*inC*outC`.

Now on to computational cost. For a regular convolution, we perform `F*F*inC`

operations at each position of the input (to compute the 2D convolution over 3

dimensions). For the whole input, the number of computations is thus

`F*F*inC*S*S` and taking all the output channels we get `F*F*inC*S*S*outC`.

For depthwise separable convolutions we need `F*F*inC*S*S*` operations for

the depthwise part; then we need `S*S*inC*outC` operations for the mixing

part. Let’s use some real numbers to get a feel for the difference:

We’ll assume `S=128`, `F=3`, `inC=3`, `outC=16`. For regular

convolution:

- Parameters:
`3*3*3*16 = 432` - Computation cost:
`3*3*3*128*128*16 = ~7e6`

For depthwise separable convolution:

- Parameters:
`3*3*3+3*16 = 75` - Computation cost:
`3*3*3*128*128+128*128*3*16 = ~1.2e6`

[1] | The term separable comes from image processing, wherespatially separable convolutions are sometimes used to save oncomputation resources. A spatial convolution is separable when the 2D convolution filter can be expressed as an outer product of two vectors. This lets us compute some 2D convolutions more cheaply. In the case of DNNs, the spatial filter is not necessarily separable but the channel dimension is separable from the spatial dimensions. |