Motivated by objects such as electric fields or fluid streams, we study the problem of learning stochastic fields, i.e. stochastic processes whose samples are fields like those occurring in physics and engineering. Considering general transformations such as rotations and reflections, we show that spatial invariance of stochastic fields requires an inference model to be equivariant. Leveraging recent advances from the equivariance literature, we study equivariance in two classes of models. Firstly, we fully characterise equivariant Gaussian processes. Secondly, we introduce Steerable Conditional Neural Processes (SteerCNPs), a new, fully equivariant member of the Neural Process family. In experiments with Gaussian process vector fields, images, and real-world weather data, we observe that SteerCNPs significantly improve the performance of previous models and equivariance leads to improvements in transfer learning tasks.
Department of Mathematics and Mathematical Statistics（数学与统计学系）；Umeå University（乌梅大学）；Department of Mathematical Sciences（数学科学系）；Chalmers University of Technology and Gothenburg University（楚姆勒技术大学和哥德堡大学）
University of Amsterdam（阿姆斯特丹大学）；EPFL（洛桑联邦理工学院）；CIFAR（加拿大高等研究院）；Qualcomm AI Research（高通人工智能研究院）；University of Copenhagen（哥本哈根大学）
We present a convolutional network that is equivariant to rigid body motions. The model uses scalar-, vector-, and tensor fields over 3D Euclidean space to represent data, and equivariant convolutions to map between such representations. These SE(3)-equivariant convolutions utilize kernels which are parameterized as a linear combination of a complete steerable kernel basis, which is derived analytically in this paper. We prove that equivariant convolutions are the most general equivariant linear maps between fields over R^3. Our experimental results confirm the effectiveness of 3D Steerable CN…
UvA-Bosch Delta Lab（阿姆斯特丹大学-博世德尔塔实验室）；University of Amsterdam（阿姆斯特丹大学）
The effectiveness of Convolutional Neural Networks (CNNs) has been substantially attributed to their built-in property of translation equivariance. However, CNNs do not have embedded mechanisms to handle other types of transformations. In this work, we pay attention to scale changes, which regularly appear in various tasks due to the changing distances between the objects and the camera. First, we introduce the general theory for building scale-equivariant convolutional networks with steerable filters. We develop scale-convolution and generalize other common blocks to be scale-equivariant. We …
University of Amsterdam（阿姆斯特丹大学）；Johannes Kepler University Linz（林茨约翰内斯·开普勒大学）；UvA-Bosch DeltaLab（阿姆斯特丹大学-博世德尔塔实验室）
Including covariant information, such as position, force, velocity or spin is important in many tasks in computational physics and chemistry. We introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks, such that node and edge attributes are not restricted to invariant scalars, but can contain covariant information, such as vectors or tensors. This model, composed of steerable MLPs, is able to incorporate geometric and physical information in both the message and update functions. Through the definition of steerable node attributes, the MLP…
Johannes-Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学）
Invariance under symmetry is an important problem in machine learning. Our paper looks specifically at equivariant neural networks where transformations of inputs yield homomorphic transformations of outputs. Here, steerable CNNs have emerged as the standard solution. An inherent problem of steerable representations is that general nonlinear layers break equivariance, thus restricting architectural choices. Our paper applies harmonic distortion analysis to illuminate the effect of nonlinearities on Fourier representations of SO(2). We develop a novel FFT-based algorithm for computing represent…
The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of $E(2)$-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations…
Convolutional neural networks revolutionized computer vision and natrual language processing. Their efficiency, as compared to fully connected neural networks, has its origin in the architecture, where convolutions reflect the translation invariance in space and time in pattern or speech recognition tasks. Recently, Cohen and Welling have put this in the broader perspective of invariance under symmetry groups, which leads to the concept of group equivaiant neural networks and more generally steerable neural networks. In this article, we review the architecture of such networks including equiva…
University of Amsterdam（阿姆斯特丹大学）；University of Heidelberg（海德堡大学）
In many machine learning tasks it is desirable that a model's prediction transforms in an equivariant way under transformations of its input. Convolutional neural networks (CNNs) implement translational equivariance by construction; for other transformations, however, they are compelled to learn the proper mapping. In this work, we develop Steerable Filter CNNs (SFCNNs) which achieve joint equivariance under translations and rotations by design. The proposed architecture employs steerable filters to efficiently compute orientation dependent responses for many orientations without suffering int…
EPFL（洛桑联邦理工学院）；Massachusetts Institute of Technology（麻省理工学院）；Microsoft Research AI4Science（微软研究院AI4Science）；University of Oxford（牛津大学）
Fragment-based drug discovery has been an effective paradigm in early-stage drug development. An open challenge in this area is designing linkers between disconnected molecular fragments of interest to obtain chemically-relevant candidate drug molecules. In this work, we propose DiffLinker, an E(3)-equivariant 3D-conditional diffusion model for molecular linker design. Given a set of disconnected fragments, our model places missing atoms in between and designs a molecule incorporating all the initial fragments. Unlike previous approaches that are only able to connect pairs of molecular fragmen…
Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are usually defined by convolutions with a kernel, whereas they are partial differential operators (PDOs) in physics. Developing the theory of equivariant PDOs in the context of deep learning could bring these subjects even closer together and lead to a stronger flow of ideas. In this work, we derive a $G$-steerability constraint that completely characterizes when a …
Qualcomm Research, Qualcomm Technologies Netherlands B.V.（高通研究，高通技术荷兰有限公司）；EPFL（洛桑联邦理工学院）；University of Amsterdam（阿姆斯特丹大学）
Group equivariant and steerable convolutional neural networks (regular and steerable G-CNNs) have recently emerged as a very effective model class for learning from signal data such as 2D and 3D images, video, and other data where symmetries are present. In geometrical terms, regular G-CNNs represent data in terms of scalar fields ("feature channels"), whereas the steerable G-CNN can also use vector or tensor fields ("capsules") to represent data. In algebraic terms, the feature spaces in regular G-CNNs transform according to a regular representation of the group G, whereas the feature spaces …
We propose an algorithm for learning a conditional generative model of a molecule given a target. Specifically, given a receptor molecule that one wishes to bind to, the conditional model generates candidate ligand molecules that may bind to it. The distribution should be invariant to rigid body transformations that act $\textit{jointly}$ on the ligand and the receptor; it should also be invariant to permutations of either the ligand or receptor atoms. Our learning algorithm is based on a continuous normalizing flow. We establish semi-equivariance conditions on the flow which guarantee the afo…
Inverted AI（Inverted AI公司）；University of British Columbia（不列颠哥伦比亚大学）；Mila（米拉研究所）
We present a novel, conditional generative probabilistic model of set-valued data with a tractable log density. This model is a continuous normalizing flow governed by permutation equivariant dynamics. These dynamics are driven by a learnable per-set-element term and pairwise interactions, both parametrized by deep neural networks. We illustrate the utility of this model via applications including (1) complex traffic scene generation conditioned on visually specified map information, and (2) object bounding box generation conditioned directly on images. We train our model by maximizing the exp…
Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This wo…
Stanford University（斯坦福大学）；Ecole Polytechnique, IP Paris（巴黎理工学院综合理工学校）
A wide range of techniques have been proposed in recent years for designing neural networks for 3D data that are equivariant under rotation and translation of the input. Most approaches for equivariance under the Euclidean group $\mathrm{SE}(3)$ of rotations and translations fall within one of the two major categories. The first category consists of methods that use $\mathrm{SE}(3)$-convolution which generalizes classical $\mathbb{R}^3$-convolution on signals over $\mathrm{SE}(3)$. Alternatively, it is possible to use \textit{steerable convolution} which achieves $\mathrm{SE}(3)$-equivariance …
Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）；Tsinghua University（清华大学）；Peking University（北京大学）
Deep learning has achieved tremendous success in designing novel chemical compounds with desirable pharmaceutical properties. In this work, we focus on a new type of drug design problem -- generating a small "linker" to physically attach two independent molecules with their distinct functions. The main computational challenges include: 1) the generation of linkers is conditional on the two given molecules, in contrast to generating full molecules from scratch in previous works; 2) linkers heavily depend on the anchor atoms of the two molecules to be connected, which are not known beforehand; 3…
Emerging from low-level vision theory, steerable filters found their counterpart in prior work on steerable convolutional neural networks equivariant to rigid transformations. In our work, we propose a steerable feed-forward learning-based approach that consists of neurons with spherical decision surfaces and operates on point clouds. Such spherical neurons are obtained by conformal embedding of Euclidean space and have recently been revisited in the context of learning representations of point sets. Focusing on 3D geometry, we exploit the isometry property of spherical neurons and derive a 3D…
Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学耿丽人工智能学院）；Beijing Key Laboratory of Big Data Management and Analysis Methods（北京大数据管理与分析方法重点实验室）；Laboratories, Huawei Technologies, Shanghai（华为技术有限公司上海2012实验室）
Qualcomm AI Research（高通人工智能研究院）；New York University（纽约大学）
Continuous input signals like images and time series that are irregularly sampled or have missing values are challenging for existing deep learning methods. Coherently defined feature representations must depend on the values in unobserved regions of the input. Drawing from the work in probabilistic numerics, we propose Probabilistic Numeric Convolutional Neural Networks which represent features as Gaussian processes (GPs), providing a probabilistic description of discretization error. We then define a convolutional layer as the evolution of a PDE defined on this GP, followed by a nonlinearity…
University of Amsterdam（阿姆斯特丹大学）；AMLab；Informatics Institute（信息学研究所）；AI4Science Lab（AI4Science实验室）；Anton Pannekoek Institute for Astronomy（安东·潘内科克天文研究所）；Microsoft Research（微软研究院）；ELLIS Unit Linz（林茨ELLIS单元）；Institute for Machine Learning（机器学习研究所）；JKU Linz（约翰内斯·开普勒大学林茨分校）；NXAI GmbH（NXAI有限公司）
Scalable Peptide Design via Memory-Efficient Equivariant Transformer
Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）；Institute for AI Industry Research, Tsinghua University（清华大学人工智能产业研究院）；Department of Electronic Engineering, Tsinghua University（清华大学电子工程系）；Department of Chemistry, Tsinghua University（清华大学化学系）
Recent improvements in generative adversarial visual synthesis incorporate real and fake image transformation in a self-supervised setting, leading to increased stability and perceptual fidelity. However, these approaches typically involve image augmentations via additional regularizers in the GAN objective and thus spend valuable network capacity towards approximating transformation equivariance instead of their desired task. In this work, we explicitly incorporate inductive symmetry priors into the network architectures via group-equivariant convolutional networks. Group-convolutions have hi…
Steerable Conditional Diffusion for Out-of-Distribution Adaptation in Medical Image Reconstruction
University College London（伦敦大学学院）；Atinary Technologies（Atinary科技公司）；University of Bremen（不来梅大学）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Ajou University School of Medicine（亚洲大学医学院）；The Chinese University of Hong Kong（香港中文大学）
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Max Planck Institute for Gravitational Physics（马克斯·普朗克引力物理研究所）；University of Tübingen（图宾根大学）
Equivariant Diffusion for Crystal Structure Prediction
School of Computer Science；Engineering, Sun Yat-sen University, Guangzhou, China；National Supercomputer Center in Guangzhou, China；Dept. of Comp. Sci. \& Tech., Institute for AI, Tsinghua University, Beijing, China；Institute for AIR, Tsinghua University, Beijing, China；Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China；Beijing Key Laboratory of Big Data Management；Analysis Methods, Beijing, China
Leveraging SO(3)-steerable convolutions for pose-robust semantic segmentation in 3D medical data
Support Center for Advanced Neuroimaging (SCAN), University Institute of Diagnostic and Interventional Neuroradiology, University of Bern, Inselspital, Bern University Hospital（高级神经影像支持中心（SCAN）、伯尔尼大学诊断与介入神经放射学研究所、伯尔尼大学、因塞尔斯皮塔尔医院、伯尔尼大学医院）；NVIDIA（英伟达）
Group equivariant neural networks have been explored in the past few years and are interesting from theoretical and practical standpoints. They leverage concepts from group representation theory, non-commutative harmonic analysis and differential geometry that do not often appear in machine learning. In practice, they have been shown to reduce sample and model complexity, notably in challenging tasks where input transformations such as arbitrary rotations are present. We begin this work with an exposition of group representation theory and the machinery necessary to define and evaluate integra…
The University of Tokyo（东京大学）；RIKEN AIP（理化学研究所人工智能项目中心）
We present the group equivariant conditional neural process (EquivCNP), a meta-learning method with permutation invariance in a data set as in conventional conditional neural processes (CNPs), and it also has transformation equivariance in data space. Incorporating group equivariance, such as rotation and scaling equivariance, provides a way to consider the symmetry of real-world data. We give a decomposition theorem for permutation-invariant and group-equivariant maps, which leads us to construct EquivCNPs with an infinite-dimensional latent space to handle group symmetries. In this paper, we…
Bayesian 3D Steerable CNNs: Enabling Equivariance and Uncertainty Quantification Simultaneously
Michigan Technological University（密歇根理工大学）；Johns Hopkins University（约翰霍普金斯大学）；The Center for Artificial Intelligence at the Institute of Computing and Cybersystems, Michigan Technological University（密歇根理工大学计算与网络系统研究所人工智能中心）
Theoretical uncertainty limits our ability to extract cosmological information from baryonic fields such as the thermal Sunyaev-Zel'dovich (tSZ) effect. Being sourced by the electron pressure field, the tSZ effect depends on baryonic physics that is usually modeled by expensive hydrodynamic simulations. We train neural networks on the IllustrisTNG-300 cosmological simulation to predict the continuous electron pressure field in galaxy clusters from gravity-only simulations. Modeling clusters is challenging for neural networks as most of the gas pressure is concentrated in a handful of voxels an…
University Of Cambridge（剑桥大学）；Secondmind Labs（Secondmind实验室）；University Of Oxford（牛津大学）；Center for Science of Data, ENS Ulm（巴黎高等师范学院乌尔姆校区数据科学中心）；Microsoft Research（微软研究院）
We present e3nn, a generalized framework for creating E(3) equivariant trainable functions, also known as Euclidean neural networks. e3nn naturally operates on geometry and geometric tensors that describe systems in 3D and transform predictably under a change of coordinate system. The core of e3nn are equivariant operations such as the TensorProduct class or the spherical harmonics functions that can be composed to create more complex modules such as convolutions and attention mechanisms. These core operations of e3nn can be used to efficiently articulate Tensor Field Networks, 3D Steerable CN…
Structure-based Drug Design with Equivariant Diffusion Models
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of Cambridge（剑桥大学）；Cornell University（康奈尔大学）；Prescient Design, Genentech（普雷赛特设计公司，基因泰克公司）；Chinese Academy of Mathematics and System Science（中国科学院数学与系统科学研究院）；Heart and Lung Research Institute, University of Cambridge（剑桥大学心肺研究所）；University of Rome “La Sapienza”（罗马大学）；Microsoft Research AI4Science（微软研究院AI4Science）；University of Amsterdam（阿姆斯特丹大学）；University of Oxford（牛津大学）；AITHYRA Institute（艾西拉研究所）
