Bronstein M. Michael, Bruna Joan, LeCun Yann, Szlam Arthur, Vanderheynst Pierre, Geometric Deep Learning

The article, IEEE Signal Processing Magazine, Digital Object Identifier, 11.07.2017, 18 - 42 p.Many scientific fields study data with an underlying structure that is non-Euclidean. 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 as 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 article is to overview different examples of geometric deep-learning problems and presently available solutions, key difficulties, applications, and future research directions in this nascent field.Overview of deep learning.
Going beyond Euclidian data.
Geometric learning problems.
- Structure of the domain;
- Data on a domain;
- Brief history;
- Signal processing, differential geometry, and graph theory;
Deep learning on Euclidian domains.
- CNN
The geometry of manifolds and graphs.
- Manifolds;
- Calculus on manifolds;
- Graphs and discrete differential operators;
- Discrete manifolds;
- Fourier analysis on non-Euclidian domains;
- Uniqueness and stability;
Spectral methods.
- Spectral CNN;
- Spectral CNN with smooth spectral multipliers;
Spectrum-free methods.
- GCNN, also known as ChebNet;
- Graph convolutional network;
- GNN;
Charting-based methods.
- Geodesic CNN;
- Anisotropic CNN;
- Mixture model network;
Combined spatial/spectral methods.
- Windowed Fourier transform;
- Wavelets;
- Localized SCNN;
Application.
- Network analysis;
- Recommender systems;
- Computer vision and graphics;
- Particle physics and chemistry;
- Molecule design;
- Medical imaging;
Open problems and future directions.
- Generalization;
- Time-varying domains;
- Directed graphs;
- Synthesis problems;
- Computation;