Posit AI Weblog: Implementing rotation equivariance: Group-equivariant CNN from scratch

Convolutional neural networks (CNNs) are nice – they’re in a position to detect options in a picture regardless of the place. Properly, not precisely. They’re not detached to simply 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 manner 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 sort 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 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 submit 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 observe a pocket book supplied 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 clarify the final pondering, and the way the ensuing structure is constructed up from smaller modules, every of which is assigned a transparent function. 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 right this moment, gcnn implements one symmetry group: (C_4), the one which serves as a working 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 identification, and know the way 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 parts’ motion. Implementation-wise, we have to distinguish between them performing 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 carried out 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 parts, not scalar tensors.

Issues are a bit much less simple the place the group motion on (mathbb{R}^2) is anxious. Right here, we’d like 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’ve an enter sign, a tensor we’d wish 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 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 report their peak. One choice 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 return down, and after they’re again, we measure their peak. The consequence is identical: Physique peak is equivariant (greater than that: invariant, even) to the motion of working up or down. (In fact, peak is a fairly boring measure. However one thing extra attention-grabbing, comparable to 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’s one matrix for every group ingredient. For (C_4), the so-called commonplace 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 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 to be 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 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 seems to be 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 attainable. Concretely, we need to use nn_conv2d(). What we’d like, 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 attainable 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 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 converse, 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 appreciate 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 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 is still to be executed 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 to be like every previous 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 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 nicely. A closing linear layer will then present the requested classifier output (of dimension out_channels).

And there now we have 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 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 “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’ll carry out considerably higher than the shift-equivariant-only commonplace 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,
  prepare = 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?

test_images <- coro::accumulate(
  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 prepare a traditional CNN. It’s as just 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 = 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) >
       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) 

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 could be a good consequence! Let’s wrap up right this 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 relatively, the samples of digits displayed, we discover an issue. In row two, column 4, there’s a digit that “beneath regular circumstances”, ought 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 can ask: does this have to be an issue? Perhaps 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 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:

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 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, because of the viewing angle – made up of squares (or edges, should you like). Now, for such a fence, varieties of rotation equivariance comparable to that encoded by (C_4) make quite a lot 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 prime of the hierarchy.

Thanks for studying!

Photograph by Marjan Blan | @marjanblan on Unsplash