Naturally Occurring Equivariance in Neural Networks

Equivariance in Neural Networks

Convolutional neural networks contain a hidden world of symmetries within themselves. This symmetry is a powerful tool in understanding the features and circuits inside neural networks. It also suggests that efforts to design neural networks with additional symmetries baked in (e.g. ) may be on a promising track.

To see these symmetries, we need to look at the individual neurons inside convolutional neural networks and the circuits that connect them. It turns out that many neurons are slightly transformed versions of the same basic feature. This includes rotated copies of the same feature, scaled copies, flipped copies, features detecting different colors, and much more. We sometimes call this phenomenon “equivariance,” since it means that switching the neurons is equivalent to transforming the input.

Before we talk about the examples introduced in this article, let’s talk about how this definition maps to the classic example of equivariance in neural networks: translation and convolutional neural network nets. In a conv net, translating the input image is equivalent to translating the neurons in the hidden layers (ignoring pooling, striding, etc). Formally, $g\in Z^2$

Now let’s consider the case of curve detectors (the first example in the Equivariant Features section), which have ten rotated copies. In this case, $g\in Z_{10}$

Conclusion

Equivariance has a remarkable ability to simplify our understanding of neural networks. When we see neural networks as families of features, interacting in structured ways, understanding small templates can actually turn into understanding how large numbers of neurons interact. Equivariance is a big help whenever we discover it.

We sometimes think of understanding neural networks as being like reverse engineering a regular computer program. In this analogy, equivariance is like finding the same inlined function repeated throughout the code. Once you realize that you’re seeing many copies of the same function, you only need to understand it once.

But natural equivariance does have some limitations. For starters, we have to find the equivariant families. This can actually take us quite a bit of work, poring through neurons. Further, they may not be exactly equivariant: one unit may be wired up slightly differently, or have a small exception, and so understanding it as equivariant could leave gaps in our understanding.

FAQs

Q: What is equivariance in neural networks?
A: Equivariance in neural networks refers to the ability of a network to preserve the symmetry of its input data under certain transformations, such as rotations, translations, or scaling.

Q: Why is equivariance important in neural networks?
A: Equivariance is important because it allows us to simplify our understanding of neural networks by identifying patterns and symmetries that exist within the data.