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

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.

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

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

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 methods 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 performing on one another, and their motion on the vector area (mathbb{R}^2), the place our enter photos dwell. The previous half is the straightforward one: It might merely be carried out by including angles. In actual fact, 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 easy 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 prefer to function on in a roundabout way. (That “a way” shall 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 prefer to document their peak. One choice we’ve got is to take the measurement, then allow them to run up. Our measurement shall be as legitimate up the mountain because it was down right here. Alternatively, we is likely to be 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 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 fascinating, equivalent 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 factor. 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 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()

A goat sitting comfortably on a meadow.

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(
    checklist(
      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()

Same goat, rotated up by 90 degrees.

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 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 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 a further dimension to understand the product of translations and rotations, the output shouldn’t be four-, however five-dimensional.

Step 3: Group convolutions

Now that we’re in “group-extended area”, 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 is still to be finished is package deal 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 like every 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 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 nicely. A closing linear layer will then present the requested classifier output (of dimension out_channels).

And there we’ve got 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 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 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 would 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_random_rotation(
        levels = c(0, 360),
        resample = 2,
        fill = 0
      )
  
)

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)
  })

32 digits, rotated randomly.

We first outline and prepare 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 = 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::torch_squeeze() 
 )

fitted <- default_cnn |>
    luz::setup(
      loss = torch::nn_cross_entropy_loss(),
      optimizer = torch::optim_adam,
      metrics = checklist(
        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 shouldn’t 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 = checklist(
      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 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 relatively, 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, 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.) Nonetheless, you can ask: does this have to be an issue? Perhaps the community simply must study the subtleties, the sorts of issues a human would spot?

The best way I view it, all of it is determined by the context: What actually ought 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 truthful, simply machine studying preserve reminding us of:

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

In our case, although, there’s 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 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 inform you why I selected the goat image. The goat is seen by way of a red-and-white fence, a sample – barely rotated, as a result of viewing angle – made up of squares (or edges, in the event you like). Now, for such a fence, sorts of rotation equivariance equivalent to that encoded by (C_4) make lots of sense. The goat itself, although, we’d relatively 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 process is use relatively versatile layers on the backside, and more and more restrained layers on the high of the hierarchy.

Thanks for studying!

Photograph by Marjan Blan | @marjanblan on Unsplash

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