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 irrespective of the place. Properly, not precisely. They’re not detached to only any sort 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 means spherical). If we would like “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 sort of motion is not going to solely register the moved function per se, but additionally, maintain observe of which concrete motion made it seem the place it’s.

That is the second publish 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. For those who haven’t, please check out that publish 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 supplied 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 overall pondering, and the way the ensuing structure is constructed up from smaller modules, every of which is assigned a transparent goal. For that motive, I received’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 have a look at the code.

As of at this time, gcnn implements one symmetry group: (C_4), the one which serves as a working instance all through publish 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.

# 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} ]

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 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 house (mathbb{R}^2), the place our enter pictures reside. The previous half is the simple one: It could merely be applied by including angles. In truth, 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 worried. Right here, we want the idea of a group illustration. That is an concerned subject, which we received’t go into right here. In our present context, it really works about like this: We’ve got an enter sign, a tensor we’d wish to function on in a roundabout way. (That “a way” might 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 choice we have now is to take the measurement, then allow them to run up. Our measurement might 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 return down, and once 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 working up or down. (After all, peak is a reasonably uninteresting measure. However one thing extra attention-grabbing, akin to coronary heart price, 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 normal 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 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 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])
    )
))

# Remodel 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 wish to make use of current, environment friendly torch performance as a lot as doable. Concretely, we wish to use nn_conv2d(). What we want, although, is a convolution kernel that’s equivariant not simply to translation, but additionally 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 reworked grid.

Why, although, name this a lifting convolution? The same old convolution kernel operates on (mathbb{R}^2); whereas our prolonged model operates on mixtures 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 understand 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 are able to chain quite a lot 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 performed 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 several 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 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 last linear layer will then present the requested classifier output (of dimension out_channels).

And there we have now 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 “excellent” – 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 can carry out considerably higher than the shift-equivariant-only normal structure.

First, we put together the info; 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 = 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::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 = 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$avg_pool() 
 )

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 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 = 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 check and coaching units are rather a lot 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 publish.

A problem

Going again to the augmented check set, or quite, the samples of digits displayed, we discover an issue. In row two, column 4, there’s a digit that “beneath regular circumstances”, needs 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 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 dependent upon the context: What actually needs to be achieved, and the way an utility 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 maintain reminding us of:

At all times consider the best way 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 by way of a red-and-white fence, a sample – barely rotated, because of the viewing angle – made up of squares (or edges, in case you like). Now, for such a fence, kinds of rotation equivariance akin to that encoded by (C_4) make loads of sense. The goat itself, although, we’d quite 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 job is use quite 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