What is geometric deep learning?

Recent advancement in deep learning has revolutionized the world of technology and has equipped it with the power to make impactful decisions. In deep learning, data is represented in Euclidean space as a 2d grid or vectors. However, we are now moving towards data being represented in the non-euclidean space. The non-euclidean space consists of curved geometries, where the straight line distance is not always the shortest. An example of the data in non-euclidean space is shown in the illustration below.

Geometric deep learning

In geometric deep learning, we enable a neural network to handle the unique properties of graphs, manifolds, or 3D structures and to capture the connectivity patterns. It analyzes the underlying and hidden patterns of the geometric structure of the data. The architecture of the geometric neural network is given below.

Mathematical intuition

The graph represents the data as nodes and edges in geometric networks. Consider a graph $G(V,E)$, where $V$ represents the set of nodes and represents the set of edges. We need to learn a model that can extract meaningful information.

For this purpose, we use the graph convolutional operation. The equation for the graph convolutional network is given below.

$h_v^l = σ(∑_{u∈N(v)}(θ^l x_u^l) + b^l)$

Here,

• $h_v^l$ is the hidden representation of node $v$ at layer $l$.

• $σ$ is an activation function.

• $N(v)$ represents the set of neighboring nodes of $v$.

• $θ^l$ is the weight matrix at layer $l$.

• $x_u^l$ is the feature vector of node $u$ at layer $l$.

• $b^l$ represents the bias term at layer $l$.

This equation shows how the representation of neighboring nodes helps in computing the hidden representation of the node.

Example

Let us take an example where each node represents an individual, and the edges of the node represent an individual's friendship. In this case, we need a graph neural network to predict whether the person will adopt the behavior of their friends based on the influence and relationship of individuals.

Moreover, we can make a geometric model to predict the 3D shape. Each point in the shape act as a node, and the edges represent the connection with neighbor nodes. By propagating the information with nearby nodes, we can extract meaningful information.

Application of geometric deep learning

Geometric deep learning has a wide range of applications in various fields. Some essential and notable applications of geometric deep learning are as follows:

• Geometric deep learning helps in estimations, and it helps classify 3D objects in computer vision tasks.

• It enables the robots to understand the spatial information from the environment by governing the structure of an object.

• It is used to predict the disease by observing the shape and structure of an organ in a non-euclidean space.

• In augmented reality, geometric transformations like rotation, translation, or scaling can be applied to augment the training data and improve model robustness.

Conclusion

Geometric deep learning has opened a new world of technology where machines can interact with 3D models. This technology has enhanced autonomous technology, allowing machines to understand complex data and infer the relation between the variables in a dataset. Geometric deep learning has enhanced augmented and virtual realities, which are based in a 3D virtual world. Furthermore, it has impacted blockchain because geometric deep learning aids in federating learning approaches in which GDL models are trained collaboratively on decentralized devices.