Explain Geometric Deep Learning.

The article Geometric deep learning: going beyond Euclidean data (by Michael M. Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, Pierre Vandergheynst) provides an overview of this relatively new sub-field of deep learning. It answers all the questions asked above (and more). If you are familiar with deep learning, graphs, linear algebra and calculus, you should be able to follow this article.

What is geometric deep learning (GDL)? This article describes GDL as follows Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. So, the inputs to these GDL models are graphs (or representations of graphs), or, in general, any non-Euclidean data. To be more concrete, the input to these models (e.g. graph neural networks) are e.g. feature vectors associated with the nodes of the graphs and matrices which describe the graph structure (e.g. the adjacency matrix of the graphs).

Why are e.g. graphs non-Euclidean data?

A graph is a non-Euclidean structure because e.g. distances between nodes are not well defined. Yes, you can have graphs with weights associated with the edges, but not all graphs have this property. What classes of problems does GDL address? In GDL, there are two classes of problems that are often tackled: characterise the structure of the data (e.g. of a graph) analyse functions defined on a given non-Euclidean domain

These classes of problems are related, given that the structure of the graph imposes certain properties on the functions that can be defined on it. Furthermore, these properties of these functions can also convey information about the structure of the graph. What are applications of GDL?

An example of an application where this type of data (graphs) arises is in the context of social networks, where each user can be associated with a vertex of the social graph and the characteristics (or features) of each user (e.g. number of friends) can be represented as a feature vector (which can then be associated with the corresponding vertex of a graph). In this context, the goal might e.g. be to determine different groups of users in the social network (i.e. clustering).

Why can't we simply use deep learning methods (like CNNs) when the data is non-Euclidean? There are several problems that arise when dealing with non-Euclidean data. For example, operations like convolution are not (usually) defined on non-Euclidean data. More concretely, the relative position of nodes is not defined on graphs (but this would be required to perform the usual convolution operation): in other words, it is meaningless to talk about a vertex that is e.g. on the left of another vertex. In practice, it means that we can't simply use the usual CNN when we are given non-Euclidean data. There have been attempts to generalise the convolution operation to graphs (or to approximate it). The field is still quite new, so there will certainly be new developments and breakthroughs.