Convolutional neural networks (CNNs) are nice – they’re in a position to detect options in a picture irrespective of the place. Nicely, not precisely. They’re not detached to simply any sort 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 observe of which concrete motion made it seem the place it’s.

**That is the second put up 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. If you happen to haven’t, please check out that put up first, since right here I’ll make use of terminology and ideas it launched.

At present, 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 will’t be thanked sufficient for making out there such wonderful studying supplies.

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

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

As of in the present day, `gcnn`

implements one symmetry group: (C_4), the one which serves as a working instance all through put up 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 are able to 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 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 photos reside. 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 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: We have now an enter sign, a tensor we’d wish to function on not directly. (That “a way” can 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 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 wish to report their top. One possibility we have now is to take the measurement, then allow them to run up. Our measurement can 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 after they’re again, we measure their top. The outcome is identical: Physique top is equivariant (greater than that: invariant, even) to the motion of working up or down. (After all, top is a fairly boring measure. However one thing extra fascinating, reminiscent of 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 *common* 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 technique’s code seems 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(
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 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 remodeled grid. The goat now seems as much as the sky.

## Step 2: The lifting convolution

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

performance as a lot as potential. Concretely, we wish 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 potential rotation.

Implementing that concept is precisely 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 remodeled 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 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 will chain quite a few 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 is still to be achieved is package deal this up. That’s what `gcnn::GroupEquivariantCNN()`

does.

## Step 4: Group-equivariant CNN

We are able to 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 like all 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 effectively. A ultimate linear layer will then present the requested classifier output (of dimension `out_channels`

).

And there we have now the entire structure. It’s time for a real-world(*ish*) take a look at.

## Rotated digits!

The concept 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’s going to carry out considerably higher than the shift-equivariant-only commonplace structure.

First, we put together the information; specifically, 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) >
self$conv1()
)
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)
```

```
Prepare 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 prepare 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)
```

```
Prepare 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 could be a good outcome! Let’s wrap up in the present day’s exploit resuming a thought from the primary, extra high-level put up.

## A problem

Going again to the augmented take a look at set, or relatively, the samples of digits displayed, we discover an issue. In row two, column 4, there’s a digit that “below regular circumstances”, must 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 can 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 best way I view it, all of it relies on the context: What actually must be achieved, and the way an utility goes for use. With digits on a letter, I’d see no cause 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 hold reminding us of:

All the time consider the way in which an utility goes for use!

In our case, although, there may be one other side 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 via 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, varieties of rotation equivariance reminiscent of that encoded by (C_4) make loads of sense. The goat itself, although, we’d relatively 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 relatively 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