Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data

By Marc Finzi, Samuel Stanton, Pavel Izmailov, Andrew Gordon Wilson. 2020

396 citations recorded by the source corpus.

The translation equivariance of convolutional layers enables convolutional neural networks to generalize well on image problems. While translation equivariance provides a powerful inductive bias for images, we often additionally desire equivariance to other transformations, such as rotations, especially for non-image data. We propose a general method to construct a convolutional layer that is equivariant to transformations from any specified Lie group with a surjective exponential map. Incorporating equivariance to a new group requires implementing only the group exponential and logarithm maps, enabling rapid prototyping. Showcasing the simplicity and generality of our method, we apply the same model architecture to images, ball-and-stick molecular data, and Hamiltonian dynamical systems. For Hamiltonian systems, the equivariance of our models is especially impactful, leading to exact conservation of linear and angular momentum.

Subfields: Artificial Intelligence, Materials Chemistry, Molecular Biology, Model Architectures

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Materials Chemistry

In Molecular Biology

In Model Architectures

Citations in the corpus

E(n) Equivariant Graph Neural Networks (2021, methodology): “Finzi et al. (2020) parametrize transformations by mapping kernels on the Lie Algebra.”
Group Equivariant Stand-Alone Self-Attention For Vision (2020, methodology): “A way to detach weights from particular kernel positions comes by parametrizing convolutional kernels as (constrained) neural networks (Thomas et al., 2018; Finzi et al., 2020; Fuchs et al., 2020).”
SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks (2020, neutral): “026 LieConv(T3) [8] .”
Inductive biases for deep learning of higher-level cognition (2020, background): “, 2020), invariance or equivariance to known transformations (Bruna et al., 2013; Defferrard et al., 2016; Ravanbakhsh et al., 2017; Thomas et al., 2018; Finzi et al., 2020; Satorras et al., 2021) or choices of prior distributions in a Bayesian model (Jeffreys, 1946; Berger and Bernardo,”
Learning local equivariant representations for large-scale atomistic dynamics (2022, neutral): “6 Cormorant [23] 22 21 21 20 LieConv [55] 19 19 24 22 L1Net [56] 13.”
Scientific discovery in the age of artificial intelligence (2023, background): “…sense, geometric deep learning involves discovering relational patterns 65 and equipping neural network models with inductive biases that explicitly make use of localized information encoded in the form of graphs and transformation groups 77–79 through neural message-passing algorithms 80–84 .”
Equivariant Diffusion for Molecule Generation in 3D (2022, background): “Leveraging these symmetries in molecular data is important for good generalization and has been extensively studied (Thomas et al., 2018; Fuchs et al., 2020; Finzi et al., 2020).”
A Survey of Deep Learning for Scientific Discovery (2020, background): “…span an extremely large area of research, so there are some areas, e.g. deep neural networks for set structured data [262, 113], modelling diﬀerent invariances — invariances to speciﬁed Lie groups for application to molecular property prediction [58], spherical invariances [35, 36] not covered.”
SE(3)-Equivariant Energy-based Models for End-to-End Protein Folding (2021, methodology): “Regular representations are used in [7, 10, 15] as an alternative approach for achieving the SE(3)-equivariance.”
GemNet: Universal Directional Graph Neural Networks for Molecules (2021, background): “They were then transferred to the context of 3D point clouds and molecules by Anderson et al. [1], Thomas et al. [56], Weiler et al. [58], and further developed by Batzner et al. [4], Finzi et al. [24], Fuchs et al. [26].”
Wavelet Networks: Scale Equivariant Learning From Raw Waveforms (2020, background): “[20] Marc Finzi, Samuel Stanton, Pavel Izmailov, and Andrew Gordon Wilson.”
Geometric and Physical Quantities improve E(3) Equivariant Message Passing (2021, methodology): “Hence, such kernels can be expanded in a continuous basis such as spherical harmonics (Weiler et al., 2018), B-splines (Bekkers, 2019) or they can be parametrised via MLPs (Finzi et al., 2020).”
Discovering Symbolic Models from Deep Learning with Inductive Biases (2020, background): “GNs can also be embedded with additional symmetries as in [23, 24], but we do not implement these.”
Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Docking (2021, methodology): “Motivated by symmetries naturally occurring in different data types, architectures are tailored to explicitly incorporate such properties (Cohen and Welling, 2016a;b; Thomas et al., 2018; Fuchs et al., 2020; Finzi et al., 2020; Eismann et al., 2020; Satorras et al., 2021).”
MetaSDF: Meta-learning Signed Distance Functions (2020, background): “Convolutional encoders, while fast, require observations on a regular grid, and are not equivariant to 3D transformations [3].”