Geometric deep learning: going beyond Euclidean data (1611.08097v2)

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.

• The paper introduces novel methods to generalize convolutional neural networks on non-Euclidean domains, enhancing the processing of graph and manifold data.
• It leverages spectral, spectrum-free, and charting-based approaches to efficiently capture complex geometric structures.
• The work demonstrates broad applicability in fields such as network analysis, computer vision, and particle physics, paving the way for future innovations.

Geometric Deep Learning: Going Beyond Euclidean Data

Introduction

The paper of geometric deep learning sits at the intersection of multiple fields of mathematics and computer science. The paper addresses the challenges and methodologies associated with generalizing deep learning models to non-Euclidean domains such as graphs and manifolds. Traditional deep learning models, particularly convolutional neural networks (CNNs), have shown significant success in processing data with a Euclidean structure, such as images and video. However, many scientific fields encounter types of data that naturally reside in non-Euclidean spaces, necessitating the development of new techniques and models.

Geometric Learning Problems

The authors classify geometric learning problems into two broad categories. The first involves characterizing the structure of the data, commonly encountered in manifold learning and non-linear dimensionality reduction. The second deals with analyzing functions that are defined over a given non-Euclidean domain. These two classes are inherently related because understanding the properties of functions on a domain provides insights into the domain itself and vice versa.

For manifold learning, classic methods like Laplacian eigenmaps and diffusion maps are extended by techniques like graphlets and motif-based representations to comprehend graph structures. For analyzing data on a domain, kernel-based methods and spectral approaches are used to generalize deep learning models such as CNNs to graph settings.

Spectral Methods

A significant portion of this paper discusses spectral methods as a means to generalize CNNs. The spectral approach starts by leveraging the eigendecomposition of the Laplacian matrix associated with a non-Euclidean domain. This matrix plays a crucial role similar to the Fourier basis in classical settings. Spectral CNNs, such as those proposed by Bruna et al., define convolutional layers in the frequency domain. Despite their conceptual simplicity, these methods face challenges regarding computational efficiency and generalizability across different domains.

Smooth spectral multipliers, as discussed by Henaff et al., reduce overfitting risks by focusing on localized filters. The polynomial parameterization approach, championed by Defferrard et al. (ChebNet) and the Graph Convolutional Network (GCN) by Kipf and Welling, offer computational efficiencies by approximating spectral filters with polynomial expressions.

Spectrum-Free Methods

The authors highlight methodologies that avoid explicit computation of the Laplacian eigenbasis, referred to as spectrum-free methods. Graph Neural Networks (GNNs) indicate a broad class where trainable parameters adapt linear combinations of the graph's low- and high-pass operators. Such models, including ChebNet and GCN, use polynomial filters that act on the neighborhood of each vertex, ensuring computational efficiency and allowing better generalization.

Charting-Based Methods

The paper also explores charting-based methods, which construct local patches around each point in a non-Euclidean domain, effectively generalizing classic Euclidean CNN operations. Geodesic CNNs and Anisotropic CNNs utilize localized coordinate systems and geometric properties, such as intrinsic distances and anisotropic diffusion kernels, to achieve this generalization. The Mixture Model Network (MoNet) further extends this idea by learning the local patch operators, combining flexibility with robust performance across various applications.

Combined Spatial/Spectral Methods

Combining spatial and spectral techniques offers a hybrid approach that leverages the strengths of both. The Windowed Fourier Transform and wavelets are employed to design convolution-like operations that are localized both in space and frequency. These combined methods strike a balance between capturing global structures and maintaining spatial localization—an essential property for many geometric applications.

Applications and Implications

The practical implications of geometric deep learning span multiple domains:

1. Network Analysis: Graph Neural Networks and related models improve tasks such as node classification and community detection in graphs.
2. Recommender Systems: Multi-Graph Convolutional Neural Networks (MGCNNs) have demonstrated success in addressing matrix completion problems in recommendation systems by considering the underlying user and item graphs.
3. Computer Vision and Graphics: Geometric deep learning provides deformable shape understanding and robust 3D shape recognition, crucial for applications in computer graphics and vision.
4. Particle Physics and Chemistry: Geometric models offer a data-driven alternative to traditional physics-based methods by predicting molecular properties and classifying particle collisions in high-energy physics.
5. Medical Imaging: Functional brain networks can be efficiently analyzed using graph-based methods to detect neurological abnormalities.

Future Directions

Several key challenges remain. There is a need for methods that can generalize across dynamically changing domains, such as evolving networks or dynamic shapes captured in 3D video. Directed graphs, commonly encountered in citation and social networks, present specific challenges due to the asymmetry in their Laplacian matrices. Another emerging area is synthesis problems, where generating data on non-Euclidean domains opens new research avenues.

Finally, while the mathematical underpinnings of geometric deep learning are well-established, computational efficiencies and the applicability of modern GPU architectures to these models need to be further explored.

Conclusion

Geometric deep learning expands the horizon of traditional deep learning into complex domains encountered in various scientific fields. By leveraging spectral, spatial, and hybrid methods, researchers are making significant strides in understanding and applying machine learning to non-Euclidean data. The journey is ongoing, and future innovations are likely to be driven by the unique challenges that each novel application domain presents.