Geometric deep learning: going beyond Euclidean data

TL;DR

This work surveys geometric deep learning as a framework to extend deep nets to non-Euclidean data such as graphs and manifolds, outlining problem classes, methods, challenges, and applications. It categorizes approaches into spectral, spectrum-free, charting-based, and combined spatial/spectral paradigms, linking them to foundations in differential geometry and graph theory. The authors provide a structured taxonomy, discuss advantages and limitations of each paradigm, and illustrate applications across networks, graphics, vision, and science. They also identify open problems and future directions, emphasizing generalization, dynamic domains, and scalable computation for geometric data.

Abstract

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. 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. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field.

Figures (13)

• Figure 1: Top: tangent space and tangent vectors on a two-dimensional manifold (surface). Bottom: Examples of isometric deformations.
• Figure 2: A toy example illustrating the difficulty of generalizing spectral filtering across non-Euclidean domains. Left: a function defined on a manifold (function values are represented by color); middle: result of the application of an edge-detection filter in the frequency domain; right: the same filter applied on the same function but on a different (nearly-isometric) domain produces a completely different result. The reason for this behavior is that the Fourier basis is domain-dependent, and the filter coefficients learnt on one domain cannot be applied to another one in a straightforward manner.
• Figure 3: Top: examples of intrinsic weighting functions used to construct a patch operator at the point marked in black (different colors represent different weighting functions). Diffusion distance (left) allows to map neighbor points according to their distance from the reference point, thus defining a one-dimensional system of local intrinsic coordinates. Anisotropic heat kernels (middle) of different scale and orientations and geodesic polar weights (right) are two-dimensional systems of coordinates. Bottom: representation of the weighting functions in the local polar $(\rho,\theta)$ system of coordinates (red curves represent the 0.5 level set).
• Figure 4: Geometric matrix completion exemplified on the famous Netflix movie recommendation problem. The column and row graphs represent the relationships between users and items, respectively.
• Figure 5: Illustration of the difference between classical CNN (left) applied to a 3D shape (checkered surface) considered as a Euclidean object, and a geometric CNN (right) applied intrinsically on the surface. In the latter case, the convolutional filters (visualized as a colored window) are deformation-invariant by construction.