Geometric deep learning (GDL) addresses two main challenges in classical deep learning: the absence of a global reference frame and the curse of high dimensionality. It leverages low-dimensional geometric priors and an associated symmetry group (𝔾), such as translations, rotations, and permutations, under which certain properties of an object remain invariant.

The properties of a protein structure (object 𝑜) defined in Euclidean space (domain Ω), such as the binding affinity between two interactors, remain invariant under roto-translation transformations from *SE(3)* (group 𝔾). We use GDL to design neural networks 𝑓 that conserve these properties.

Let's set the stage with a few key definitions:

**Domain **Ω (e.g. 3D Euclidean space) is an underlying space where we can define an object. We apply function space 𝕆 on domain Ω to define object 𝑜 (e.g. a protein 3D structure).

where

is the range of values taken by objects defined on Ω.

**Group** 𝔾 is a set equipped with a binary operation ⊗, known as composition, and satisfies the following properties:

Closure:

Associativity:

Identity:

Inverse:

Each transformation *g *∈𝔾 has a representation 𝑝(*g*). For example, if *g* is a translation in Euclidean space then,

is a translation matrix.

If domain Ω is symmetrical under group 𝔾 then it ensures that the properties of the object 𝑜 remain unchanged despite transformations defined within 𝔾. The Euclidean space is endowed with roto-translation symmetries and the group that contains the roto-translation transformations is called the special Euclidean group or 𝑆𝐸(𝑛), where n is the number of dimensions. This gives us *inductive bias* which eventually leads to the definition of invariant and equivariant functions, the building blocks of GDL.

*Equivariant block*

The function 𝑓1 (e.g. a neural network) is equivariant under the transformation *g* if applying *g* to the input object 𝑜 results in the same transformation being applied to the output. GDL employs patch-wise symmetry groups to achieve this and ensures that the entire architecture remains locally equivariant by stacking multiple locally equivariant layers.

where 𝑓1 is a function defined on the object to extract its properties and is equivariant to the group 𝔾.

*Invariant block*

We can formulate invariant function 𝑓2 (again a neural network) that benefits from the geometric priors of domain Ω:

where 𝑓2 is a function defined on the object to extract its properties and is invariant to the group 𝔾.

In most problems, the final layer is designed to make the whole process ** globally** invariant. The integration of equivariant and invariant blocks minimizes the number of trainable parameters by utilizing kernels with shared weights (a simple example is convolutional neural networks) and removes the necessity for data augmentation.

Now let's see how we can put together a complete GDL pipeline!

If domain Ω' is a compact (coarse-grained) version of domain Ω (Ω′ ⊆ Ω), then we can define the building blocks of the GDL approaches:

*Linear *𝔾-*equivariant layer*

𝔾*-invariant layer (global pooling)*

*Local pooling (coarsening) *

*Nonlinearity*

Now we simply concatenate these blocks to create 𝔾-invariant function:

Two standout GDL frameworks that incorporate equivariant and invariant blocks are Graph Neural Networks (GNNs) and Group Equivariant Convolutional Networks (G-CNN). These are widely used in computational and structural biology for many tasks, including quality assessment of protein structure models, protein structure prediction, protein interface prediction, and protein design.

I highly recommend the resources below. Back in 2021, I enjoyed reading [1]! It was so well-explained and easy to follow - I learned a lot from it.

Here are also some amazing resources from Amsterdam Machine Learning Lab (AMLab) and the University of Amsterdam,

Hands-on notebooks: __GDL - Regular Group Convolutions__ and __Group Equivariant Neural Networks__.

Lectures on __Group Equivariant Deep Learning__

This blog is also nice: