Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2104.13478v2)

Abstract: The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

• The paper introduces a unifying framework that leverages geometric invariances to design diverse deep learning architectures.
• It presents detailed methodologies for CNNs, GNNs, and group-equivariant networks by exploiting translation and rotational symmetries.
• The study demonstrates that incorporating geometric priors enhances efficiency, robustness, and interpretability in practical AI applications.

Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges

"Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges" is a comprehensive exploration of modern deep learning architectures unified through geometric and symmetry-based considerations. This work systematically analyzes various neural network architectures, including convolutional neural networks (CNNs), graph neural networks (GNNs), and others, emphasizing their reliance on geometric principles and invariances.

Core Concepts

The paper introduces Geometric Deep Learning as a discipline where different deep learning architectures can be understood and designed based on principles of symmetries and invariances. These principles allow neural networks to respect the geometric structure of the data they process, leading to more efficient, interpretable, and generalizable models.

Geometric Priors

The fundamental idea is to exploit known geometric priors of the data domain:

• Symmetry: Ensuring that the network's output is invariant or equivariant to certain transformations of the input.
• Scale Separation: Decomposition of input data into multiscale representations to handle variability at different levels effectively.

Domains and Architectures

The research categorizes neural networks into domains:

• Grids: Traditional CNNs operate on grid-structured data and leverage translation symmetry via convolutions.
• Groups: Extending convolutional notions to data domains acted on by more complex symmetry groups (e.g., spherical CNNs).
• Graphs: GNNs handle data represented as graphs, ensuring permutation invariance and leveraging local neighborhoods for node feature aggregation.
• Geodesics and Manifolds: Considering smooth manifolds where data possesses intrinsic geometric structure, such as meshes used in 3D shape analysis.
• Gauges: Dealing with structured vector bundles where local frames or gauges must be preserved or transformed consistently under certain conditions.

Key Neural Architectures

Convolutional Neural Networks

CNNs utilize convolutions to ensure translation equivariance and parameter sharing, crucially including pooling layers for hierarchical feature extraction and scale separation. Residual networks (ResNets) improve training depth by introducing identity mappings, interpreted as discretized differential operators.

Group-equivariant CNNs

The extension of CNNs to more complex symmetries involves group convolutions enabling architectures to be invariant or equivariant to groups beyond translations, such as rotations and other orthogonal transformations. This is particularly useful for data with inherent rotational symmetries like spherical data.

Graph Neural Networks

GNNs generalize convolutions to graphs by local neighborhood aggregation, employing methods such as message passing, attention mechanisms, and even spectral methods derived from graph signal processing. Key flavors include:

• Convolutional GNNs: Fixed-weight aggregation over neighborhoods.
• Attentional GNNs: Learning adaptive weights for aggregation.
• Message Passing GNNs: Computing and aggregating more complex messages across edges.

Geometric Variants

Equivariant Message Passing Networks

Addressing the need for equivariance to specific transformations (e.g., Euclidean transformations in 3D), these networks extend GNNs by ensuring that geometric transformations applied to input data are consistently reflected in the output features.

Intrinsic Mesh CNNs

For meshes representing discrete manifolds, operations such as geodesic convolutions respect the intrinsic geometry of the data. Gauge-equivariant filters can handle local coordinate systems' rotations, providing a robust framework for processing non-Euclidean surfaces.

Applications and Implications

Geometric Deep Learning has broad applications, from 3D shape analysis in computer graphics to molecular property prediction in computational chemistry, enhancing both theoretical understanding and practical performance. The paper suggests that future advancements in AI will focus on better leveraging these geometric and symmetry-based priors, leading to more powerful and generalizable architectures.

Conclusions

The paper serves as a unifying framework for diverse deep learning models through geometric principles, which helps in systematic design and understanding of neural networks. This approach not only yields performance improvements but also promotes interpretability, robustness, and theoretical rigor in model development. Future work will undoubtedly build on this foundation by exploring new domains and invariances, further bridging the gap between mathematical theory and practical AI applications.