Convolutional neural networks (CNNs) are nice – they’re capable of detect options in a picture irrespective of the place. Nicely, not precisely. They’re not detached to only any sort of motion. Shifting up or down, or left or proper, is okay; rotating round an axis shouldn’t be. That’s due to how convolution works: traverse by row, then traverse by column (or the opposite method spherical). If we would like “extra” (e.g., profitable detection of an upside-down object), we have to lengthen convolution to an operation that’s *rotation-equivariant*. An operation that’s *equivariant* to some kind of motion is not going to solely register the moved characteristic per se, but additionally, preserve monitor of which concrete motion made it seem the place it’s.

**That is the second submit in a sequence that introduces group-equivariant CNNs (GCNNs)***.* The first was a high-level introduction to why we’d need them, and the way they work. There, we launched the important thing participant, the symmetry group, which specifies what sorts of transformations are to be handled equivariantly. When you haven’t, please check out that submit first, since right here I’ll make use of terminology and ideas it launched.

Right now, we code a easy GCNN from scratch. Code and presentation tightly observe a notebook supplied as a part of College of Amsterdam’s 2022 Deep Learning Course. They will’t be thanked sufficient for making out there such glorious studying supplies.

In what follows, my intent is to clarify the final pondering, and the way the ensuing structure is constructed up from smaller modules, every of which is assigned a transparent goal. For that motive, I received’t reproduce all of the code right here; as an alternative, I’ll make use of the package deal `gcnn`

. Its strategies are closely annotated; so to see some particulars, don’t hesitate to have a look at the code.

As of at this time, `gcnn`

implements one symmetry group: (C_4), the one which serves as a operating instance all through submit one. It’s straightforwardly extensible, although, making use of sophistication hierarchies all through.

## Step 1: The symmetry group (C_4)

In coding a GCNN, the very first thing we have to present is an implementation of the symmetry group we’d like to make use of. Right here, it’s (C_4), the four-element group that rotates by 90 levels.

We will ask `gcnn`

to create one for us, and examine its components.

```
torch_tensor
0.0000
1.5708
3.1416
4.7124
[ CPUFloatType{4} ]
```

Parts are represented by their respective rotation angles: (0), (frac{pi}{2}), (pi), and (frac{3 pi}{2}).

Teams are conscious of the id, and know learn how to assemble a component’s inverse:

```
C_4$id
g1 <- elems[2]
C_4$inverse(g1)
```

```
torch_tensor
0
[ CPUFloatType{1} ]
torch_tensor
4.71239
[ CPUFloatType{} ]
```

Right here, what we care about most is the group components’ *motion*. Implementation-wise, we have to distinguish between them appearing on one another, and their motion on the vector area (mathbb{R}^2), the place our enter photographs stay. The previous half is the simple one: It might merely be applied by including angles. In actual fact, that is what `gcnn`

does after we ask it to let `g1`

act on `g2`

:

```
g2 <- elems[3]
# in C_4$left_action_on_H(), H stands for the symmetry group
C_4$left_action_on_H(torch_tensor(g1)$unsqueeze(1), torch_tensor(g2)$unsqueeze(1))
```

```
torch_tensor
4.7124
[ CPUFloatType{1,1} ]
```

What’s with the `unsqueeze()`

s? Since (C_4)’s final *raison d’être* is to be a part of a neural community, `left_action_on_H()`

works with batches of components, not scalar tensors.

Issues are a bit much less simple the place the group motion on (mathbb{R}^2) is worried. Right here, we want the idea of a group representation. That is an concerned subject, which we received’t go into right here. In our present context, it really works about like this: We have now an enter sign, a tensor we’d prefer to function on not directly. (That “a way” will probably be convolution, as we’ll see quickly.) To render that operation group-equivariant, we first have the illustration apply the *inverse* group motion to the enter. That completed, we go on with the operation as if nothing had occurred.

To present a concrete instance, let’s say the operation is a measurement. Think about a runner, standing on the foot of some mountain path, able to run up the climb. We’d prefer to document their peak. One choice we’ve got is to take the measurement, then allow them to run up. Our measurement will probably be as legitimate up the mountain because it was down right here. Alternatively, we may be well mannered and never make them wait. As soon as they’re up there, we ask them to come back down, and after they’re again, we measure their peak. The consequence is similar: Physique peak is equivariant (greater than that: invariant, even) to the motion of operating up or down. (In fact, peak is a fairly boring measure. However one thing extra fascinating, reminiscent of coronary heart charge, wouldn’t have labored so nicely on this instance.)

Returning to the implementation, it seems that group actions are encoded as matrices. There may be one matrix for every group ingredient. For (C_4), the so-called *normal* illustration is a rotation matrix:

[

begin{bmatrix} cos(theta) & -sin(theta) sin(theta) & cos(theta) end{bmatrix}

]

In `gcnn`

, the operate making use of that matrix is `left_action_on_R2()`

. Like its sibling, it’s designed to work with batches (of group components in addition to (mathbb{R}^2) vectors). Technically, what it does is rotate the grid the picture is outlined on, after which, re-sample the picture. To make this extra concrete, that methodology’s code appears about as follows.

Here’s a goat.

```
img_path <- system.file("imgs", "z.jpg", package deal = "gcnn")
img <- torchvision::base_loader(img_path) |> torchvision::transform_to_tensor()
img$permute(c(2, 3, 1)) |> as.array() |> as.raster() |> plot()
```

First, we name `C_4$left_action_on_R2()`

to rotate the grid.

```
# Grid form is [2, 1024, 1024], for a second, 1024 x 1024 picture.
img_grid_R2 <- torch::torch_stack(torch::torch_meshgrid(
list(
torch::torch_linspace(-1, 1, dim(img)[2]),
torch::torch_linspace(-1, 1, dim(img)[3])
)
))
# Rework the picture grid with the matrix illustration of some group ingredient.
transformed_grid <- C_4$left_action_on_R2(C_4$inverse(g1)$unsqueeze(1), img_grid_R2)
```

Second, we re-sample the picture on the reworked grid. The goat now appears as much as the sky.

## Step 2: The lifting convolution

We need to make use of current, environment friendly `torch`

performance as a lot as attainable. Concretely, we need to use `nn_conv2d()`

. What we want, although, is a convolution kernel that’s equivariant not simply to translation, but additionally to the motion of (C_4). This may be achieved by having one kernel for every attainable rotation.

Implementing that concept is precisely what `LiftingConvolution`

does. The precept is similar as earlier than: First, the grid is rotated, after which, the kernel (weight matrix) is re-sampled to the reworked grid.

Why, although, name this a *lifting convolution*? The same old convolution kernel operates on (mathbb{R}^2); whereas our prolonged model operates on combos of (mathbb{R}^2) and (C_4). In math communicate, it has been *lifted* to the semi-direct product (mathbb{R}^2rtimes C_4).

```
lifting_conv <- LiftingConvolution(
group = CyclicGroup(order = 4),
kernel_size = 5,
in_channels = 3,
out_channels = 8
)
x <- torch::torch_randn(c(2, 3, 32, 32))
y <- lifting_conv(x)
y$form
```

`[1] 2 8 4 28 28`

Since, internally, `LiftingConvolution`

makes use of an extra dimension to comprehend the product of translations and rotations, the output shouldn’t be four-, however five-dimensional.

## Step 3: Group convolutions

Now that we’re in “group-extended area”, we are able to chain plenty of layers the place each enter and output are *group convolution* layers. For instance:

```
group_conv <- GroupConvolution(
group = CyclicGroup(order = 4),
kernel_size = 5,
in_channels = 8,
out_channels = 16
)
z <- group_conv(y)
z$form
```

`[1] 2 16 4 24 24`

All that continues to be to be completed is package deal this up. That’s what `gcnn::GroupEquivariantCNN()`

does.

## Step 4: Group-equivariant CNN

We will name `GroupEquivariantCNN()`

like so.

```
cnn <- GroupEquivariantCNN(
group = CyclicGroup(order = 4),
kernel_size = 5,
in_channels = 1,
out_channels = 1,
num_hidden = 2, # variety of group convolutions
hidden_channels = 16 # variety of channels per group conv layer
)
img <- torch::torch_randn(c(4, 1, 32, 32))
cnn(img)$form
```

`[1] 4 1`

At informal look, this `GroupEquivariantCNN`

appears like every outdated CNN … weren’t it for the `group`

argument.

Now, after we examine its output, we see that the extra dimension is gone. That’s as a result of after a sequence of group-to-group convolution layers, the module tasks all the way down to a illustration that, for every batch merchandise, retains channels solely. It thus averages not simply over areas – as we usually do – however over the group dimension as nicely. A closing linear layer will then present the requested classifier output (of dimension `out_channels`

).

And there we’ve got the whole structure. It’s time for a real-world(*ish*) take a look at.

## Rotated digits!

The thought is to coach two convnets, a “regular” CNN and a group-equivariant one, on the same old MNIST coaching set. Then, each are evaluated on an augmented take a look at set the place every picture is randomly rotated by a steady rotation between 0 and 360 levels. We don’t anticipate `GroupEquivariantCNN`

to be “good” – not if we equip with (C_4) as a symmetry group. Strictly, with (C_4), equivariance extends over 4 positions solely. However we do hope it should carry out considerably higher than the shift-equivariant-only normal structure.

First, we put together the information; particularly, the augmented take a look at set.

```
dir <- "/tmp/mnist"
train_ds <- torchvision::mnist_dataset(
dir,
obtain = TRUE,
rework = torchvision::transform_to_tensor
)
test_ds <- torchvision::mnist_dataset(
dir,
practice = FALSE,
rework = operate(x) >
torchvision::transform_to_tensor()
)
train_dl <- dataloader(train_ds, batch_size = 128, shuffle = TRUE)
test_dl <- dataloader(test_ds, batch_size = 128)
```

How does it look?

We first outline and practice a traditional CNN. It’s as much like `GroupEquivariantCNN()`

, architecture-wise, as attainable, and is given twice the variety of hidden channels, in order to have comparable capability total.

```
default_cnn <- nn_module(
"default_cnn",
initialize = operate(kernel_size, in_channels, out_channels, num_hidden, hidden_channels) {
self$conv1 <- torch::nn_conv2d(in_channels, hidden_channels, kernel_size)
self$convs <- torch::nn_module_list()
for (i in 1:num_hidden) {
self$convs$append(torch::nn_conv2d(hidden_channels, hidden_channels, kernel_size))
}
self$avg_pool <- torch::nn_adaptive_avg_pool2d(1)
self$final_linear <- torch::nn_linear(hidden_channels, out_channels)
},
ahead = operate(x) >
self$avg_pool()
)
fitted <- default_cnn |>
luz::setup(
loss = torch::nn_cross_entropy_loss(),
optimizer = torch::optim_adam,
metrics = list(
luz::luz_metric_accuracy()
)
) |>
luz::set_hparams(
kernel_size = 5,
in_channels = 1,
out_channels = 10,
num_hidden = 4,
hidden_channels = 32
) %>%
luz::set_opt_hparams(lr = 1e-2, weight_decay = 1e-4) |>
luz::match(train_dl, epochs = 10, valid_data = test_dl)
```

```
Practice metrics: Loss: 0.0498 - Acc: 0.9843
Legitimate metrics: Loss: 3.2445 - Acc: 0.4479
```

Unsurprisingly, accuracy on the take a look at set shouldn’t be that nice.

Subsequent, we practice the group-equivariant model.

```
fitted <- GroupEquivariantCNN |>
luz::setup(
loss = torch::nn_cross_entropy_loss(),
optimizer = torch::optim_adam,
metrics = list(
luz::luz_metric_accuracy()
)
) |>
luz::set_hparams(
group = CyclicGroup(order = 4),
kernel_size = 5,
in_channels = 1,
out_channels = 10,
num_hidden = 4,
hidden_channels = 16
) |>
luz::set_opt_hparams(lr = 1e-2, weight_decay = 1e-4) |>
luz::match(train_dl, epochs = 10, valid_data = test_dl)
```

```
Practice metrics: Loss: 0.1102 - Acc: 0.9667
Legitimate metrics: Loss: 0.4969 - Acc: 0.8549
```

For the group-equivariant CNN, accuracies on take a look at and coaching units are rather a lot nearer. That could be a good consequence! Let’s wrap up at this time’s exploit resuming a thought from the primary, extra high-level submit.

## A problem

Going again to the augmented take a look at set, or quite, the samples of digits displayed, we discover an issue. In row two, column 4, there’s a digit that “underneath regular circumstances”, needs to be a 9, however, most likely, is an upside-down 6. (To a human, what suggests that is the squiggle-like factor that appears to be discovered extra usually with sixes than with nines.) Nonetheless, you could possibly ask: does this *have* to be an issue? Possibly the community simply must study the subtleties, the sorts of issues a human would spot?

The way in which I view it, all of it is dependent upon the context: What actually needs to be completed, and the way an utility goes for use. With digits on a letter, I’d see no motive why a single digit ought to seem upside-down; accordingly, full rotation equivariance can be counter-productive. In a nutshell, we arrive on the identical canonical crucial advocates of truthful, simply machine studying preserve reminding us of:

At all times consider the way in which an utility goes for use!

In our case, although, there’s one other facet to this, a technical one. `gcnn::GroupEquivariantCNN()`

is an easy wrapper, in that its layers all make use of the identical symmetry group. In precept, there is no such thing as a want to do that. With extra coding effort, completely different teams can be utilized relying on a layer’s place within the feature-detection hierarchy.

Right here, let me lastly inform you why I selected the goat image. The goat is seen by a red-and-white fence, a sample – barely rotated, as a result of viewing angle – made up of squares (or edges, should you like). Now, for such a fence, sorts of rotation equivariance reminiscent of that encoded by (C_4) make a variety of sense. The goat itself, although, we’d quite not have look as much as the sky, the way in which I illustrated (C_4) motion earlier than. Thus, what we’d do in a real-world image-classification process is use quite versatile layers on the backside, and more and more restrained layers on the high of the hierarchy.

Thanks for studying!

Picture by Marjan Blan | @marjanblan on Unsplash