Convolutional neural networks (CNNs) are nice – they’re in a position to detect options in a picture irrespective of the place. Effectively, not precisely. They’re not detached to only any type of motion. Shifting up or down, or left or proper, is okay; rotating round an axis will not be. That’s due to how convolution works: traverse by row, then traverse by column (or the opposite method spherical). If we wish “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 sort of motion is not going to solely register the moved characteristic per se, but additionally, hold monitor of which concrete motion made it seem the place it’s.

**That is the second submit in a collection 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. In case you haven’t, please check out that submit first, since right here I’ll make use of terminology and ideas it launched.

As we speak, we code a easy GCNN from scratch. Code and presentation tightly observe a pocket book offered as a part of College of Amsterdam’s 2022 Deep Studying Course. They will’t be thanked sufficient for making out there such glorious studying supplies.

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

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

As of at the moment, `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} ]
```

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

Teams are conscious of the identification, and know find out how to assemble a component’s inverse:

```
C_4$identification
g1 <- elems[2]
C_4$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 house (mathbb{R}^2), the place our enter pictures stay. The previous half is the straightforward one: It might merely be applied by including angles. The truth is, that is what `gcnn`

does once 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 illustration. That is an concerned subject, which we gained’t go into right here. In our present context, it really works about like this: We’ve an enter sign, a tensor we’d wish to function on not directly. (That “a way” will likely 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 achieved, we go on with the operation as if nothing had occurred.

To offer 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 wish to document their peak. One possibility we’ve is to take the measurement, then allow them to run up. Our measurement will likely be as legitimate up the mountain because it was down right here. Alternatively, we is perhaps well mannered and never make them wait. As soon as they’re up there, we ask them to come back down, and once 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 reasonably uninteresting measure. However one thing extra fascinating, comparable to coronary heart charge, wouldn’t have labored so effectively on this instance.)

Returning to the implementation, it seems that group actions are encoded as matrices. There may be one matrix for every group aspect. For (C_4), the so-called *customary* 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 to be like about as follows.

Here’s a goat.

```
img_path <- system.file("imgs", "z.jpg", bundle = "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 2nd, 1024 x 1024 picture.
img_grid_R2 <- torch::torch_stack(torch::torch_meshgrid(
listing(
torch::torch_linspace(-1, 1, dim(img)[2]),
torch::torch_linspace(-1, 1, dim(img)[3])
)
))
# Remodel the picture grid with the matrix illustration of some group aspect.
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 to be like as much as the sky.

## Step 2: The lifting convolution

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

performance as a lot as potential. 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 potential 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 standard 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 will not be four-, however five-dimensional.

## Step 3: Group convolutions

Now that we’re in “group-extended house”, we will chain numerous 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 executed is bundle 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 to be like like several previous CNN … weren’t it for the `group`

argument.

Now, once 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 initiatives all the way down to a illustration that, for every batch merchandise, retains channels solely. It thus averages not simply over places – as we usually do – however over the group dimension as effectively. A last linear layer will then present the requested classifier output (of dimension `out_channels`

).

And there we’ve 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 count on `GroupEquivariantCNN`

to be “excellent” – 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 can carry out considerably higher than the shift-equivariant-only customary structure.

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

```
dir <- "/tmp/mnist"
train_ds <- torchvision::mnist_dataset(
dir,
obtain = TRUE,
remodel = torchvision::transform_to_tensor
)
test_ds <- torchvision::mnist_dataset(
dir,
prepare = FALSE,
remodel = 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 prepare a traditional CNN. It’s as much like `GroupEquivariantCNN()`

, architecture-wise, as potential, 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) >
((.) torch::nnf_layer_norm(., .$form[2:4]))()
)
fitted <- default_cnn |>
luz::setup(
loss = torch::nn_cross_entropy_loss(),
optimizer = torch::optim_adam,
metrics = listing(
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 will not be that nice.

Subsequent, we prepare the group-equivariant model.

```
fitted <- GroupEquivariantCNN |>
luz::setup(
loss = torch::nn_cross_entropy_loss(),
optimizer = torch::optim_adam,
metrics = listing(
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 so much nearer. That may be a good consequence! Let’s wrap up at the moment’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 moderately, the samples of digits displayed, we discover an issue. In row two, column 4, there’s a digit that “below 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 typically with sixes than with nines.) Nevertheless, you would 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 will depend on the context: What actually needs to be achieved, and the way an software 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 similar canonical crucial advocates of honest, simply machine studying hold reminding us of:

All the time consider the best way an software goes for use!

In our case, although, there may be one other side to this, a technical one. `gcnn::GroupEquivariantCNN()`

is a straightforward wrapper, in that its layers all make use of the identical symmetry group. In precept, there isn’t 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 way of a red-and-white fence, a sample – barely rotated, because of the viewing angle – made up of squares (or edges, if you happen to like). Now, for such a fence, forms of rotation equivariance comparable to that encoded by (C_4) make a number of sense. The goat itself, although, we’d moderately not have look as much as the sky, the best way I illustrated (C_4) motion earlier than. Thus, what we’d do in a real-world image-classification job is use moderately versatile layers on the backside, and more and more restrained layers on the prime of the hierarchy.

Thanks for studying!

Picture by Marjan Blan | @marjanblan on Unsplash