The one principle is simply stated as 'Let the kernel rotate' and we will focus in this article on how you can apply it in your architectures.

Equivariant architectures allow us to train models which are indifferent to certain group actions.

To understand what this exactly means, let us train this simple CNN model on the MNIST dataset (a dataset of handwritten digits from 0-9).

class SimpleCNN(nn.Module):

    def __init__(self):
        super(SimpleCNN, self).__init__()
        self.cl1 = nn.Conv2d(in_channels=1, out_channels=8, kernel_size=3, padding=1)
        self.max_1 = nn.MaxPool2d(kernel_size=2)
        self.cl2 = nn.Conv2d(in_channels=8, out_channels=16, kernel_size=3, padding=1)
        self.max_2 = nn.MaxPool2d(kernel_size=2)
        self.cl3 = nn.Conv2d(in_channels=16, out_channels=16, kernel_size=7)
        self.dense = nn.Linear(in_features=16, out_features=10)

    def forward(self, x: torch.Tensor):
        x = nn.functional.silu(self.cl1(x))
        x = self.max_1(x)
        x = nn.functional.silu(self.cl2(x))
        x = self.max_2(x)
        x = nn.functional.silu(self.cl3(x))
        x = x.view(len(x), -1)
        logits = self.dense(x)
        return logits

_{Table 1: Test accuracy of the SimpleCNN model}

As expected, we get over 95% accuracy on the testing dataset, but what if we rotate the image by 90 degrees? Without any countermeasures applied, the results drop to just slightly better than guessing. This model would be useless for general applications.

In contrast, let us train a similar equivariant architecture with the same number of parameters, where the group actions are exactly the 90-degree rotations.

_{Table 2: Test accuracy of the EqCNN model with the same amount of parameters as the SimpleCNN model}

The accuracy remains the same, and we did not even opt for data augmentation.

These models become even more impressive with 3D data, but we will stick with this example to explore the core idea.

In case you want to test it out for yourself, you can access all code written in both PyTorch and JAX for free under Github-Repo, and training with Docker or Podman is possible with just two commands.

Have fun!

So What is Equivariance?

Equivariant architectures guarantee stability of features under certain group actions. Groups are simple structures where group elements can be combined, reversed, or do nothing.

You can look up the formal definition on Wikipedia if you are interested.

For our purposes, you can think of a group of 90-degree rotations acting on square images. We can rotate an image by 90, 180, 270, or 360 degrees. To reverse the action, we apply a 270, 180, 90, or 0-degree rotation respectively. It is straightforward to see that we can combine, reverse, or do nothing with the group denoted as $C_4$C4​ . The image visualizes all actions on an image.

_{Figure 1: Rotated MNIST image by 90°, 180°, 270°, 360°, respectively}

Now, given an input image $x$x , our CNN model classifier $f_{\theta }$fθ​ , and an arbitrary 90-degree rotation $g$g , the equivariant property can be expressed as
$f_{\theta }(\text{rotate }x\text{ by }g)=f_{\theta }(x)$fθ​(rotate x by g)=fθ​(x)

Generally speaking, we want our image-based model to have the same outputs when rotated.

As such, equivariant models promise us architectures with baked-in symmetries. In the following section, we will see how our principle can be applied to achieve this property.

How to Make Our CNN Equivariant

The problem is the following: When the image rotates, the features rotate too. But as already hinted, we could also compute the features for each rotation upfront by rotating the kernel.
We could actually rotate the kernel, but it is much easier to rotate the feature map itself, thus avoiding interference with PyTorch's autodifferentiation algorithm altogether.

So, in code, our CNN kernel

x = nn.functional.silu(self.cl1(x))

now acts on all four rotated images:

x_0 = x
x_90 = torch.rot90(x, k=1, dims=(2, 3))
x_180 = torch.rot90(x, k=2, dims=(2, 3))
x_270 = torch.rot90(x, k=3, dims=(2, 3))

x_0 = nn.functional.silu(self.cl1(x_0))
x_90 = nn.functional.silu(self.cl1(x_90))
x_180 = nn.functional.silu(self.cl1(x_180))
x_270 = nn.functional.silu(self.cl1(x_270))

Or more compactly written as a 3D convolution:

self.cl1 = nn.Conv3d(in_channels=1, out_channels=8, kernel_size=(1, 3, 3))
...
x = torch.stack([x_0, x_90, x_180, x_270], dim=-3)
x = nn.functional.silu(self.cl1(x))

The resulting equivariant model has just a few lines more compared to the version above:

class EqCNN(nn.Module):

    def __init__(self):
        super(EqCNN, self).__init__()
        self.cl1 = nn.Conv3d(in_channels=1, out_channels=8, kernel_size=(1, 3, 3))
        self.max_1 = nn.MaxPool3d(kernel_size=(1, 2, 2))
        self.cl2 = nn.Conv3d(in_channels=8, out_channels=16, kernel_size=(1, 3, 3))
        self.max_2 = nn.MaxPool3d(kernel_size=(1, 2, 2))
        self.cl3 = nn.Conv3d(in_channels=16, out_channels=16, kernel_size=(1, 5, 5))
        self.dense = nn.Linear(in_features=16, out_features=10)

    def forward(self, x: torch.Tensor):
        x_0 = x
        x_90 = torch.rot90(x, k=1, dims=(2, 3))
        x_180 = torch.rot90(x, k=2, dims=(2, 3))
        x_270 = torch.rot90(x, k=3, dims=(2, 3))

        x = torch.stack([x_0, x_90, x_180, x_270], dim=-3)
        x = nn.functional.silu(self.cl1(x))
        x = self.max_1(x)

        x = nn.functional.silu(self.cl2(x))
        x = self.max_2(x)

        x = nn.functional.silu(self.cl3(x))

        x = x.squeeze()
        x = torch.max(x, dim=-1).values
        logits = self.dense(x)
        return logits

But why is this equivariant to rotations?
First, observe that we get four copies of each feature map at each stage. At the end of the pipeline, we combine all of them with a max operation.

This is key, the max operation is indifferent to which place the rotated version of the feature ends up in.

To understand what is happening, let us plot the feature maps after the first convolution stage.

_{Figure 2: Feature maps for all four rotations}

And now the same features after we rotate the input by 90 degrees.

_{Figure 3: Feature maps for all four rotations after the input image was rotated}

I color-coded the corresponding maps. Each feature map is shifted by one. As the final max operator computes the same result for these shifted feature maps, we obtain the same results.

In my code, I did not rotate back after the final convolution, since my kernels condense the image to a one-dimensional array. If you want to expand on this example, you would need to account for this fact.

Accounting for group actions or "kernel rotations" plays a vital role in the design of more sophisticated architectures.

Is it a Free Lunch?

No, we pay in computational speed, inductive bias, and a more complex implementation.

The latter point is somewhat solved with libraries such as E3NN, where most of the heavy math is abstracted away. Nevertheless, one needs to account for a lot during architecture design.

One superficial weakness is the 4x computational cost for computing all rotated feature layers. However, modern hardware with mass parallelization can easily counteract this load. In contrast, training a simple CNN with data augmentation would easily exceed 10x in training time. This gets even worse for 3D rotations where data augmentation would require about 500x the training amount to compensate for all possible rotations.

Overall, equivariance model design is more often than not a price worth paying if one wants stable features.

What is Next?

Equivariant model designs have exploded in recent years, and in this article, we barely scratched the surface. In fact, we did not even exploit the full $C_4$C4​ group yet. We could have used full 3D kernels. However, our model already achieves over 95% accuracy, so there is little reason to go further with this example.

Besides CNNs, researchers have successfully translated these principles to continuous groups, including $SO(2)$SO(2) (the group of all rotations in the plane) and $SE(3)$SE(3) (the group of all translations and rotations in 3D space).

In my experience, these models are absolutely mind-blowing and achieve performance, when trained from scratch, comparable to the performance of foundation models trained on multiple times larger datasets.

Let me know if you want me to write more on this topic.

Further References

In case you want a formal introduction to this topic, here is an excellent compilation of papers, covering the complete history of equivariance in Machine Learning.
AEN

I actually plan to create a deep-dive, hands-on tutorial on this topic. You can already sign up for my mailing list, and I will provide you with free versions over time, along with a direct channel for feedback and Q&A.

See you around :)