We introduce tensor field neural networks, which are locally equivariant to 3D rotations, translations, and permutations of points at every layer. 3D rotation equivariance removes the need for data augmentation to identify features in arbitrary orientations. Our network uses filters built from spherical harmonics; due to the mathematical consequences of this filter choice, each layer accepts as input (and guarantees as output) scalars, vectors, and higher-order tensors, in the geometric sense of these terms. We demonstrate the capabilities of tensor field networks with tasks in geometry, physics, and chemistry.
We develop a rotation equivariant model for generating 3D hand meshes from 2D RGB images. This guarantees that as the input image of a hand is rotated the generated mesh undergoes a corresponding rotation. Furthermore, this removes undesirable deformations in the meshes often generated by methods without rotation equivariance. By building a rotation equivariant model, through considering symmetries in the problem, we reduce the need for training on very large datasets to achieve good mesh reconstruction. The encoder takes images defined on $\mathbb{Z}^{2}$ and maps these to latent functions de…
Technion – Israel Institute of Technology（以色列理工学院）；NVIDIA Research（英伟达研究院）
Equivariance to permutations and rigid motions is an important inductive bias for various 3D learning problems. Recently it has been shown that the equivariant Tensor Field Network architecture is universal -- it can approximate any equivariant function. In this paper we suggest a much simpler architecture, prove that it enjoys the same universality guarantees and evaluate its performance on Modelnet40. The code to reproduce our experiments is available at \url{https://github.com/simpleinvariance/UniversalNetwork}
This paper proposes a set of rules to revise various neural networks for 3D point cloud processing to rotation-equivariant quaternion neural networks (REQNNs). We find that when a neural network uses quaternion features under certain conditions, the network feature naturally has the rotation-equivariance property. Rotation equivariance means that applying a specific rotation transformation to the input point cloud is equivalent to applying the same rotation transformation to all intermediate-layer quaternion features. Besides, the REQNN also ensures that the intermediate-layer features are inv…
Chalmers University of Technology（查尔姆斯理工大学）；Umeå University（于默奥大学）
In this paper, we are concerned with rotation equivariance on 2D point cloud data. We describe a particular set of functions able to approximate any continuous rotation equivariant and permutation invariant function. Based on this result, we propose a novel neural network architecture for processing 2D point clouds and we prove its universality for approximating functions exhibiting these symmetries. We also show how to extend the architecture to accept a set of 2D-2D correspondences as indata, while maintaining similar equivariance properties. Experiments are presented on the estimation of es…
SHReg: Strictly Rotation-Equivariant Point Cloud Registration via Spherical Harmonics
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；School of Artificial Intelligence, Shandong Women’s University（山东女子学院人工智能学院）
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
Extracting discriminative local features that are invariant to imaging variations is an integral part of establishing correspondences between images. In this work, we introduce a self-supervised learning framework to extract discriminative rotation-invariant descriptors using group-equivariant CNNs. Thanks to employing group-equivariant CNNs, our method effectively learns to obtain rotation-equivariant features and their orientations explicitly, without having to perform sophisticated data augmentations. The resultant features and their orientations are further processed by group aligning, a n…
Rotation equivariance is a desirable property in many practical applications such as motion forecasting and 3D perception, where it can offer benefits like sample efficiency, better generalization, and robustness to input perturbations. Vector Neurons (VN) is a recently developed framework offering a simple yet effective approach for deriving rotation-equivariant analogs of standard machine learning operations by extending one-dimensional scalar neurons to three-dimensional "vector neurons." We introduce a novel "VN-Transformer" architecture to address several shortcomings of the current VN mo…
Wuhan University（武汉大学）；The University of Hong Kong（香港大学）；Texas A&M University（得克萨斯农工大学）
In this paper, we propose a novel local descriptor-based framework, called You Only Hypothesize Once (YOHO), for the registration of two unaligned point clouds. In contrast to most existing local descriptors which rely on a fragile local reference frame to gain rotation invariance, the proposed descriptor achieves the rotation invariance by recent technologies of group equivariant feature learning, which brings more robustness to point density and noise. Meanwhile, the descriptor in YOHO also has a rotation equivariant part, which enables us to estimate the registration from just one correspon…
Tel Aviv University（特拉维夫大学）；Hebrew University of Jerusalem（希伯来大学）
We introduce an unsupervised technique for encoding point clouds into a canonical shape representation, by disentangling shape and pose. Our encoder is stable and consistent, meaning that the shape encoding is purely pose-invariant, while the extracted rotation and translation are able to semantically align different input shapes of the same class to a common canonical pose. Specifically, we design an auto-encoder based on Vector Neuron Networks, a rotation-equivariant neural network, whose layers we extend to provide translation-equivariance in addition to rotation-equivariance only. The resu…
This is an exploratory study that discovers the current image quantization (vector quantization) do not satisfy translation equivariance in the quantized space due to aliasing. Instead of focusing on anti-aliasing, we propose a simple yet effective way to achieve translation-equivariant image quantization by enforcing orthogonality among the codebook embeddings. To explore the advantages of translation-equivariant image quantization, we conduct three proof-of-concept experiments with a carefully controlled dataset: (1) text-to-image generation, where the quantized image indices are the target …
Subsampling is used in convolutional neural networks (CNNs) in the form of pooling or strided convolutions, to reduce the spatial dimensions of feature maps and to allow the receptive fields to grow exponentially with depth. However, it is known that such subsampling operations are not translation equivariant, unlike convolutions that are translation equivariant. Here, we first introduce translation equivariant subsampling/upsampling layers that can be used to construct exact translation equivariant CNNs. We then generalise these layers beyond translations to general groups, thus proposing gro…
Technical University of Munich（慕尼黑工业大学）；D’Annunzio University（邓南遮大学）
Convolutional networks are successful, but they have recently been outperformed by new neural networks that are equivariant under rotations and translations. These new networks work better because they do not struggle with learning each possible orientation of each image feature separately. So far, they have been proposed for 2D and 3D data. Here we generalize them to 6D diffusion MRI data, ensuring joint equivariance under 3D roto-translations in image space and the matching 3D rotations in $q$-space, as dictated by the image formation. Such equivariant deep learning is appropriate for diffus…
Michigan State University（密歇根州立大学）；Meta AI；Ford Motor Company（福特汽车公司）
Modern neural networks use building blocks such as convolutions that are equivariant to arbitrary 2D translations. However, these vanilla blocks are not equivariant to arbitrary 3D translations in the projective manifold. Even then, all monocular 3D detectors use vanilla blocks to obtain the 3D coordinates, a task for which the vanilla blocks are not designed for. This paper takes the first step towards convolutions equivariant to arbitrary 3D translations in the projective manifold. Since the depth is the hardest to estimate for monocular detection, this paper proposes Depth EquiVarIAnt NeTwo…
Simons Machine Learning Center（西蒙斯机器学习中心）；New York Structural Biology Center（纽约结构生物学中心）
In many imaging modalities, objects of interest can occur in a variety of locations and poses (i.e. are subject to translations and rotations in 2d or 3d), but the location and pose of an object does not change its semantics (i.e. the object's essence). That is, the specific location and rotation of an airplane in satellite imagery, or the 3d rotation of a chair in a natural image, or the rotation of a particle in a cryo-electron micrograph, do not change the intrinsic nature of those objects. Here, we consider the problem of learning semantic representations of objects that are invariant to p…
Department of Mathematics and Statistics, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校数学与统计系）；Department of Electrical and Computer Engineering, Purdue University（普渡大学电气与计算机工程系）；Department of Mathematics, Duke University（杜克大学数学系）；Department of Electrical and Computer Engineering, Duke University（杜克大学电气与计算机工程系）
Encoding the scale information explicitly into the representation learned by a convolutional neural network (CNN) is beneficial for many computer vision tasks especially when dealing with multiscale inputs. We study, in this paper, a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions across the space and the scaling group, which is shown to be both sufficient and necessary to achieve equivariance for the regular representation of the scaling-translation group ST . To reduce the model complexity and computational burden, we decompose the convolutional filters under two…
Bosch Center for Artificial Intelligence（博世人工智能中心）；Oxford University（牛津大学）；University of Amsterdam（阿姆斯特丹大学）
We introduce the SE(3)-Transformer, a variant of the self-attention module for 3D point clouds and graphs, which is equivariant under continuous 3D roto-translations. Equivariance is important to ensure stable and predictable performance in the presence of nuisance transformations of the data input. A positive corollary of equivariance is increased weight-tying within the model. The SE(3)-Transformer leverages the benefits of self-attention to operate on large point clouds and graphs with varying number of points, while guaranteeing SE(3)-equivariance for robustness. We evaluate our model on a…
We propose a semantic segmentation model that exploits rotation and reflection symmetries. We demonstrate significant gains in sample efficiency due to increased weight sharing, as well as improvements in robustness to symmetry transformations. The group equivariant CNN framework is extended for segmentation by introducing a new equivariant (G->Z2)-convolution that transforms feature maps on a group to planar feature maps. Also, equivariant transposed convolution is formulated for up-sampling in an encoder-decoder network. To demonstrate improvements in sample efficiency we evaluate on multipl…
University of Toronto（多伦多大学）；Adobe Research（奥多比研究院）
The Adam optimization algorithm has proven remarkably effective for optimization problems across machine learning and even traditional tasks in geometry processing. At the same time, the development of equivariant methods, which preserve their output under the action of rotation or some other transformation, has proven to be important for geometry problems across these domains. In this work, we observe that Adam $-$ when treated as a function that maps initial conditions to optimized results $-$ is not rotation equivariant for vector-valued parameters due to per-coordinate moment updates. This…
Institute for Theoretical Physics, TU Wien（维也纳技术大学理论物理研究所）
The rising adoption of machine learning in high energy physics and lattice field theory necessitates the re-evaluation of common methods that are widely used in computer vision, which, when applied to problems in physics, can lead to significant drawbacks in terms of performance and generalizability. One particular example for this is the use of neural network architectures that do not reflect the underlying symmetries of the given physical problem. In this work, we focus on complex scalar field theory on a two-dimensional lattice and investigate the benefits of using group equivariant convolu…
Deforestation has become a significant contributing factor to climate change and, due to this, both classifying the drivers and predicting segmentation maps of deforestation has attracted significant interest. In this work, we develop a rotation equivariant convolutional neural network model to predict the drivers and generate segmentation maps of deforestation events from Landsat 8 satellite images. This outperforms previous methods in classifying the drivers and predicting the segmentation map of deforestation, offering a 9% improvement in classification accuracy and a 7% improvement in segm…
We develop theory and software for rotation equivariant operators on scalar and vector fields, with diverse applications in simulation, optimization and machine learning. Rotation equivariance (covariance) means all fields in the system rotate together, implying spatially invariant dynamics that preserve symmetry. Extending the convolution theorems of linear time invariant systems, we theorize that linear equivariant operators are characterized by tensor field convolutions using an appropriate product between the input field and a radially symmetric kernel field. Most Green's functions and dif…
Rakuten Institute of Technology Paris（乐天技术研究院巴黎分院）
Deep neural networks have been shown to be vulnerable to adversarial examples: very small perturbations of the input having a dramatic impact on the predictions. A wealth of adversarial attacks and distance metrics to quantify the similarity between natural and adversarial images have been proposed, recently enlarging the scope of adversarial examples with geometric transformations beyond pixel-wise attacks. In this context, we investigate the robustness to adversarial attacks of new Convolutional Neural Network architectures providing equivariance to rotations. We found that rotation-equivari…
This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution la…
We propose a new model for digital pathology segmentation, based on the observation that histopathology images are inherently symmetric under rotation and reflection. Utilizing recent findings on rotation equivariant CNNs, the proposed model leverages these symmetries in a principled manner. We present a visual analysis showing improved stability on predictions, and demonstrate that exploiting rotation equivariance significantly improves tumor detection performance on a challenging lymph node metastases dataset. We further present a novel derived dataset to enable principled comparison of mach…
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…
Duke University（杜克大学）；NVIDIA Research（英伟达研究（NVIDIA Research））
Learning functions on point clouds has applications in many fields, including computer vision, computer graphics, physics, and chemistry. Recently, there has been a growing interest in neural architectures that are invariant or equivariant to all three shape-preserving transformations of point clouds: translation, rotation, and permutation. In this paper, we present a first study of the approximation power of these architectures. We first derive two sufficient conditions for an equivariant architecture to have the universal approximation property, based on a novel characterization of the space…
Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision tasks. This paper proposes to decompose the convolutional filters over joint steerable bases across the space and the group geometry simultaneously, namely a rotation-equivariant CNN with decomposed convolutional filters (RotDCF). This decomposition facilitates computing the joint convolution, which is proved to be necessary for the group equivariance. It significantly reduces the model size and compu…
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…
Tsinghua University（清华大学）；Institute for AI Industry Research (AIR), Tsinghua University（清华大学人工智能产业研究院（AIR））；Beijing Academy of Artificial Intelligence（北京人工智能研究院）；Renmin University of China（中国人民大学）
Antibody design is valuable for therapeutic usage and biological research. Existing deep-learning-based methods encounter several key issues: 1) incomplete context for Complementarity-Determining Regions (CDRs) generation; 2) incapability of capturing the entire 3D geometry of the input structure; 3) inefficient prediction of the CDR sequences in an autoregressive manner. In this paper, we propose Multi-channel Equivariant Attention Network (MEAN) to co-design 1D sequences and 3D structures of CDRs. To be specific, MEAN formulates antibody design as a conditional graph translation problem by i…
BioGeometry；Mila - Québec AI Institute（米拉-魁北克人工智能研究所）；Université de Montréal（蒙特利尔大学）；HEC Montréal（蒙特利尔高等商学院）
Proteins are macromolecules that perform essential functions in all living organisms. Designing novel proteins with specific structures and desired functions has been a long-standing challenge in the field of bioengineering. Existing approaches generate both protein sequence and structure using either autoregressive models or diffusion models, both of which suffer from high inference costs. In this paper, we propose a new approach capable of protein sequence and structure co-design, which iteratively translates both protein sequence and structure into the desired state from random initializati…
In the recent application of scientific modeling, machine learning models are largely applied to facilitate computational simulations of fluid systems. Rotation symmetry is a general property for most symmetric fluid systems. However, in general, current machine learning methods have no theoretical way to guarantee rotational symmetry. By observing an important property of contraction and rotation operation on high-order symmetric tensors, we prove that the rotation operation is preserved via tensor contraction. Based on this theoretical justification, in this paper, we introduce Rotation-Equi…
Freie Universität Berlin（柏林自由大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；EPFL（洛桑联邦理工学院）；Rice University（莱斯大学）
Equivariant neural networks (ENNs) are graph neural networks embedded in $\mathbb{R}^3$ and are well suited for predicting molecular properties. The ENN library e3nn has customizable convolutions, which can be designed to depend only on distances between points, or also on angular features, making them rotationally invariant, or equivariant, respectively. This paper studies the practical value of including angular dependencies for molecular property prediction directly via an ablation study with \texttt{e3nn} and the QM9 data set. We find that, for fixed network depth and parameter count, addi…
Shanghai Jiao Tong University（上海交通大学）；Southern University of Science and Technology（南方科技大学）；Huawei International（华为国际）
In order to prevent leaking input information from intermediate-layer features, this paper proposes a method to revise the traditional neural network into the rotation-equivariant neural network (RENN). Compared to the traditional neural network, the RENN uses d-ary vectors/tensors as features, in which each element is a d-ary number. These d-ary features can be rotated (analogous to the rotation of a d-dimensional vector) with a random angle as the encryption process. Input information is hidden in this target phase of d-ary features for attribute obfuscation. Even if attackers have obtained …
NTNU（挪威科技大学）；University of Bath（巴斯大学）；University of Cambridge（剑桥大学）
In recent years the use of convolutional layers to encode an inductive bias (translational equivariance) in neural networks has proven to be a very fruitful idea. The successes of this approach have motivated a line of research into incorporating other symmetries into deep learning methods, in the form of group equivariant convolutional neural networks. Much of this work has been focused on roto-translational symmetry of $\mathbf R^d$, but other examples are the scaling symmetry of $\mathbf R^d$ and rotational symmetry of the sphere. In this work, we demonstrate that group equivariant convolut…
University of California at Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Our universe is homogeneous and isotropic, and its perturbations obey translation and rotation symmetry. In this work we develop Translation and Rotation Equivariant Normalizing Flow (TRENF), a generative Normalizing Flow (NF) model which explicitly incorporates these symmetries, defining the data likelihood via a sequence of Fourier space-based convolutions and pixel-wise nonlinear transforms. TRENF gives direct access to the high dimensional data likelihood p(x|y) as a function of the labels y, such as cosmological parameters. In contrast to traditional analyses based on summary statistics, …
Numerical simulation is an essential tool in many areas of science and engineering, but its performance often limits application in practice or when used to explore large parameter spaces. On the other hand, surrogate deep learning models, while accelerating simulations, often exhibit poor accuracy and ability to generalise. In order to improve these two factors, we introduce REMuS-GNN, a rotation-equivariant multi-scale model for simulating continuum dynamical systems encompassing a range of length scales. REMuS-GNN is designed to predict an output vector field from an input vector field on a…
State-of-the-art deep learning systems often require large amounts of data and computation. For this reason, leveraging known or unknown structure of the data is paramount. Convolutional neural networks (CNNs) are successful examples of this principle, their defining characteristic being the shift-equivariance. By sliding a filter over the input, when the input shifts, the response shifts by the same amount, exploiting the structure of natural images where semantic content is independent of absolute pixel positions. This property is essential to the success of CNNs in audio, image and video re…
Proteins are miniature machines whose function depends on their three-dimensional (3D) structure. Determining this structure computationally remains an unsolved grand challenge. A major bottleneck involves selecting the most accurate structural model among a large pool of candidates, a task addressed in model quality assessment. Here, we present a novel deep learning approach to assess the quality of a protein model. Our network builds on a point-based representation of the atomic structure and rotation-equivariant convolutions at different levels of structural resolution. These combined aspec…
Convolutional neural networks (CNNs) constructed natively on the sphere have been developed recently and shown to be highly effective for the analysis of spherical data. While an efficient framework has been formulated, spherical CNNs are nevertheless highly computationally demanding; typically they cannot scale beyond spherical signals of thousands of pixels. We develop scattering networks constructed natively on the sphere that provide a powerful representational space for spherical data. Spherical scattering networks are computationally scalable and exhibit rotational equivariance, while th…
Equivariance is a nice property to have as it produces much more parameter efficient neural architectures and preserves the structure of the input through the feature mapping. Even though some combinations of transformations might never appear (e.g. an upright face with a horizontal nose), current equivariant architectures consider the set of all possible transformations in a transformation group when learning feature representations. Contrarily, the human visual system is able to attend to the set of relevant transformations occurring in the environment and utilizes this information to assist…
