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 only any form 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 approach spherical). If we wish “extra” (e.g., profitable detection of an upside-down object), we have to prolong 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 function per se, but in addition, hold observe 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. Should you haven’t, please check out that submit first, since right here I’ll make use of terminology and ideas it launched.

Right this moment, 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 accessible such glorious studying supplies.

In what follows, my intent is to clarify the final considering, and the way the ensuing structure is constructed up from smaller modules, every of which is assigned a transparent objective. For that purpose, 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 immediately, 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 are able to ask gcnn to create one for us, and examine its parts.

# remotes::install_github("skeydan/gcnn")
library(gcnn)
library(torch)

C_4 <- CyclicGroup(order = 4)
elems <- C_4$parts()
elems

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 id, and know 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 parts’ 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 photographs stay. The previous half is the simple one: It might merely be applied by including angles. In truth, 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 parts, 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 now have an enter sign, a tensor we’d wish to function on in a roundabout way. (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 achieved, 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 wish to report their peak. One possibility now we have 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. (After all, peak is a reasonably uninteresting measure. However one thing extra attention-grabbing, corresponding to coronary heart charge, wouldn’t have labored so properly on this instance.)

Returning to the implementation, it seems that group actions are encoded as matrices. There’s one matrix for every group factor. 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 parts 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 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(
    listing(
      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 factor.
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.

transformed_img <- torch::nnf_grid_sample(
  img$unsqueeze(1), transformed_grid,
  align_corners = TRUE, mode = "bilinear", padding_mode = "zeros"
)

transformed_img[1,..]$permute(c(2, 3, 1)) |> as.array() |> as.raster() |> plot()

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 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 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 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 a variety 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 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 several outdated 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 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 properly. A ultimate linear layer will then present the requested classifier output (of dimension out_channels).

And there now we have the whole structure. It’s time for a real-world(ish) check.

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 check 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 is going to carry out considerably higher than the shift-equivariant-only normal structure.

First, we put together the information; specifically, the augmented check 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?

test_images <- coro::acquire(
  test_dl, 1
)[[1]]$x[1:32, 1, , ] |> as.array()

par(mfrow = c(4, 8), mar = rep(0, 4), mai = rep(0, 4))
test_images |>
  purrr::array_tree(1) |>
  purrr::map(as.raster) |>
  purrr::iwalk(~ {
    plot(.x)
  })

We first outline and practice a standard 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 general.

 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$final_linear()
     x
   
 )

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) 

Prepare metrics: Loss: 0.0498 - Acc: 0.9843
Legitimate metrics: Loss: 3.2445 - Acc: 0.4479

Unsurprisingly, accuracy on the check set will not 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 = 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)

Prepare metrics: Loss: 0.1102 - Acc: 0.9667
Legitimate metrics: Loss: 0.4969 - Acc: 0.8549

For the group-equivariant CNN, accuracies on check and coaching units are quite a bit nearer. That may be a good consequence! Let’s wrap up immediately’s exploit resuming a thought from the primary, extra high-level submit.

A problem

Going again to the augmented check set, or slightly, the samples of digits displayed, we discover an issue. In row two, column 4, there’s a digit that “underneath 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 usually with sixes than with nines.) Nevertheless, you might 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 best way I view it, all of it depends upon the context: What actually ought to be achieved, and the way an utility 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 similar canonical crucial advocates of honest, simply machine studying hold reminding us of:

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

In our case, although, there’s 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 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 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, when you like). Now, for such a fence, varieties of rotation equivariance corresponding to that encoded by (C_4) make a whole lot of sense. The goat itself, although, we’d slightly 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 activity is use slightly 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