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 form of motion. Shifting up or down, or left or proper, is okay; rotating round an axis isn’t. That’s due to how convolution works: traverse by row, then traverse by column (or the opposite means 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 won’t solely register the moved characteristic per se, but in addition, hold 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.

In the present day, we code a easy GCNN from scratch. Code and presentation tightly comply with a pocket book offered as a part of College of Amsterdam’s 2022 Deep Studying Course. They’ll’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 goal. For that purpose, I gained’t reproduce all of the code right here; as a substitute, 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 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 identification, and know learn 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 area (mathbb{R}^2), the place our enter photographs stay. The previous half is the straightforward one: It might merely be applied by including angles. The truth is, 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 easy the place the group motion on (mathbb{R}^2) is anxious. Right here, we want the idea of a group illustration. That is an concerned matter, which we gained’t go into right here. In our present context, it really works about like this: Now we have an enter sign, a tensor we’d prefer to function on ultimately. (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 provide 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 report their peak. One possibility we’ve 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 could 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 end result is identical: Physique peak is equivariant (greater than that: invariant, even) to the motion of operating up or down. (After all, peak is a reasonably uninteresting measure. However one thing extra attention-grabbing, equivalent 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’s one matrix for every group component. For (C_4), the so-called *common* illustration is a rotation matrix:

[

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

]

In `gcnn`

, the perform 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 seems to be 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(
record(
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 component.
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 seems to be 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 doable. Concretely, we need to use `nn_conv2d()`

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

Implementing that concept is strictly what `LiftingConvolution`

does. The precept is identical 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 mixtures 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 a further dimension to comprehend the product of translations and rotations, the output isn’t four-, however five-dimensional.

## Step 3: Group convolutions

Now that we’re in “group-extended area”, we are able to 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 completed 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`

seems to be 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 initiatives right 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 ultimate linear layer will then present the requested classifier output (of dimension `out_channels`

).

And there we’ve the entire 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 standard 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 “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 would carry out considerably higher than the shift-equivariant-only normal structure.

First, we put together the info; specifically, 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 = perform(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 just like `GroupEquivariantCNN()`

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

```
default_cnn <- nn_module(
"default_cnn",
initialize = perform(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 = perform(x) >
((.) torch::nnf_layer_norm(., .$form[2:4]))()
)
fitted <- default_cnn |>
luz::setup(
loss = torch::nn_cross_entropy_loss(),
optimizer = torch::optim_adam,
metrics = record(
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 isn’t that nice.

Subsequent, we practice the group-equivariant model.

```
fitted <- GroupEquivariantCNN |>
luz::setup(
loss = torch::nn_cross_entropy_loss(),
optimizer = torch::optim_adam,
metrics = record(
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 end result! 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 slightly, the samples of digits displayed, we discover an issue. In row two, column 4, there’s a digit that “below regular circumstances”, ought to be a 9, however, 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.) Nonetheless, you might ask: does this *have* to be an issue? Possibly the community simply must be taught the subtleties, the sorts of issues a human would spot?

The way in which I view it, all of it depends upon the context: What actually ought to be completed, and the way an software goes for use. With digits on a letter, I’d see no purpose 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 honest, simply machine studying hold reminding us of:

At all times consider the best way an software 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 isn’t any 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 let you know why I selected the goat image. The goat is seen by means of a red-and-white fence, a sample – barely rotated, as a result of viewing angle – made up of squares (or edges, for those who like). Now, for such a fence, sorts of rotation equivariance equivalent to that encoded by (C_4) make plenty of sense. The goat itself, although, we’d slightly 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 activity is use slightly versatile layers on the backside, and more and more restrained layers on the prime of the hierarchy.

Thanks for studying!

Photograph by Marjan Blan | @marjanblan on Unsplash