Equivariance of neural networks to transformations helps to improve their performance and reduce generalization error in computer vision tasks, as they apply to datasets presenting symmetries (e.g. scalings, rotations, translations). The method of moving frames is classical for deriving operators invariant to the action of a Lie group in a manifold.Recently, a rotation and translation equivariant neural network for image data was proposed based on the moving frames approach. In this paper we significantly improve that approach by reducing the computation of moving frames to only one, at the input stage, instead of repeated computations at each layer. The equivariance of the resulting architecture is proved theoretically and we build a rotation and translation equivariant neural network to process volumes, i.e. signals on the 3D space. Our trained model overperforms the benchmarks in the medical volume classification of most of the tested datasets from MedMNIST3D.
Equivariant networks capture the inductive bias about the symmetry of the learning task by building those symmetries into the model. In this paper, we study how equivariance relates to generalization error utilizing PAC Bayesian analysis for equivariant networks, where the transformation laws of feature spaces are determined by group representations. By using perturbation analysis of equivariant networks in Fourier domain for each layer, we derive norm-based PAC-Bayesian generalization bounds. The bound characterizes the impact of group size, and multiplicity and degree of irreducible represen…
$G$-equivariant convolutional neural networks (GCNNs) is a geometric deep learning model for data defined on a homogeneous $G$-space $\mathcal{M}$. GCNNs are designed to respect the global symmetry in $\mathcal{M}$, thereby facilitating learning. In this paper, we analyze GCNNs on homogeneous spaces $\mathcal{M} = G/K$ in the case of unimodular Lie groups $G$ and compact subgroups $K \leq G$. We demonstrate that homogeneous vector bundles is the natural setting for GCNNs. We also use reproducing kernel Hilbert spaces to obtain a precise criterion for expressing $G$-equivariant layers as convol…
Generalization Bounds for Equivariant Networks on Markov Data
College of Mathematics, Sichuan University（四川大学数学学院）；College of Mathematics and Statistics, Sichuan University of Science and Engineering（四川轻化工大学数学与统计学院）
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Keio University（庆应义塾大学）；The University of Tokyo（东京大学）
Numerous invariant (or equivariant) neural networks have succeeded in handling invariant data such as point clouds and graphs. However, a generalization theory for the neural networks has not been well developed, because several essential factors for the theory, such as network size and margin distribution, are not deeply connected to the invariance and equivariance. In this study, we develop a novel generalization error bound for invariant and equivariant deep neural networks. To describe the effect of invariance and equivariance on generalization, we develop a notion of a \textit{quotient fe…
We propose a self-supervised framework to learn scene representations from video that are automatically delineated into objects and background. Our method relies on moving objects being equivariant with respect to their transformation across frames and the background being constant. After training, we can manipulate and render the scenes in real time to create unseen combinations of objects, transformations, and backgrounds. We show results on moving MNIST with backgrounds.
NYU（纽约大学）；NVidia（英伟达）；Technical University of Berlin（柏林工业大学）；Google（谷歌）
We propose a self-supervised framework to learn scene representations from video that are automatically delineated into background, characters, and their animations. Our method capitalizes on moving characters being equivariant with respect to their transformation across frames and the background being constant with respect to that same transformation. After training, we can manipulate image encodings in real time to create unseen combinations of the delineated components. As far as we know, we are the first method to perform unsupervised extraction and synthesis of interpretable background, c…
Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
We provide a general self-attention formulation to impose group equivariance to arbitrary symmetry groups. This is achieved by defining positional encodings that are invariant to the action of the group considered. Since the group acts on the positional encoding directly, group equivariant self-attention networks (GSA-Nets) are steerable by nature. Our experiments on vision benchmarks demonstrate consistent improvements of GSA-Nets over non-equivariant self-attention networks.
Nanjing University of Science and Technology（南京理工大学）
This paper proposes a Rotation-equivariant Attention Feature Fusion Pyramid Networks for Aerial Object Detection named ReAFFPN. ReAFFPN aims at improving the effect of rotation-equivariant features fusion between adjacent layers which suffers from the semantic and scale discontinuity. Due to the particularity of rotational equivariant convolution, general methods are unable to achieve their original effect while ensuring rotation equivariance of the network. To solve this problem, we design a new Rotation-equivariant Channel Attention which has the ability to both generate channel attention an…
Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；University of Amsterdam（阿姆斯特丹大学）
Although group convolutional networks are able to learn powerful representations based on symmetry patterns, they lack explicit means to learn meaningful relationships among them (e.g., relative positions and poses). In this paper, we present attentive group equivariant convolutions, a generalization of the group convolution, in which attention is applied during the course of convolution to accentuate meaningful symmetry combinations and suppress non-plausible, misleading ones. We indicate that prior work on visual attention can be described as special cases of our proposed framework and show …
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…
This paper proposes a method to construct pretext tasks for self-supervised learning on group equivariant neural networks. Group equivariant neural networks are the models whose structure is restricted to commute with the transformations on the input. Therefore, it is important to construct pretext tasks for self-supervised learning that do not contradict this equivariance. To ensure that training is consistent with the equivariance, we propose two concepts for self-supervised tasks: equivariant pretext labels and invariant contrastive loss. Equivariant pretext labels use a set of labels on wh…
In this paper we present a framework for the design and implementation of offset equivariant networks, that is, neural networks that preserve in their output uniform increments in the input. In a suitable color space this kind of networks achieves equivariance with respect to the photometric transformations that characterize changes in the lighting conditions. We verified the framework on three different problems: image recognition, illuminant estimation, and image inpainting. Our experiments show that the performance of offset equivariant networks are comparable to those in the state of the a…
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…
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…
Automatic lesion segmentation in dermoscopy images is an essential step for computer-aided diagnosis of melanoma. The dermoscopy images exhibits rotational and reflectional symmetry, however, this geometric property has not been encoded in the state-of-the-art convolutional neural networks based skin lesion segmentation methods. In this paper, we present a deeply supervised rotation equivariant network for skin lesion segmentation by extending the recent group rotation equivariant network~\cite{cohen2016group}. Specifically, we propose the G-upsampling and G-projection operations to adapt the …
Equivariant networks have been adopted in many 3-D learning areas. Here we identify a fundamental limitation of these networks: their ambiguity to symmetries. Equivariant networks cannot complete symmetry-dependent tasks like segmenting a left-right symmetric object into its left and right sides. We tackle this problem by adding components that resolve symmetry ambiguities while preserving rotational equivariance. We present OAVNN: Orientation Aware Vector Neuron Network, an extension of the Vector Neuron Network. OAVNN is a rotation equivariant network that is robust to planar symmetric input…
How can prior knowledge on the transformation invariances of a domain be incorporated into the architecture of a neural network? We propose Equivariant Transformers (ETs), a family of differentiable image-to-image mappings that improve the robustness of models towards pre-defined continuous transformation groups. Through the use of specially-derived canonical coordinate systems, ETs incorporate functions that are equivariant by construction with respect to these transformations. We show empirically that ETs can be flexibly composed to improve model robustness towards more complicated transform…
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 …
Recently, learning equivariant representations has attracted considerable research attention. Dieleman et al. introduce four operations which can be inserted into convolutional neural network to learn deep representations equivariant to rotation. However, feature maps should be copied and rotated four times in each layer in their approach, which causes much running time and memory overhead. In order to address this problem, we propose Deep Rotation Equivariant Network consisting of cycle layers, isotonic layers and decycle layers. Our proposed layers apply rotation transformation on filters ra…
Sri Sathya Sai Institute of Higher Learning（斯里萨提亚赛高等教育学院）
Capsule networks are constrained by the parameter-expensive nature of their layers, and the general lack of provable equivariance guarantees. We present a variation of capsule networks that aims to remedy this. We identify that learning all pair-wise part-whole relationships between capsules of successive layers is inefficient. Further, we also realise that the choice of prediction networks and the routing mechanism are both key to equivariance. Based on these, we propose an alternative framework for capsule networks that learns to projectively encode the manifold of pose-variations, termed th…
The convolutional layers of standard convolutional neural networks (CNNs) are equivariant to translation. However, the convolution and fully-connected layers are not equivariant or invariant to other affine geometric transformations. Recently, a new class of CNNs is proposed in which the conventional layers of CNNs are replaced with equivariant convolution, pooling, and batch-normalization layers. The final classification layer in equivariant neural networks is invariant to different affine geometric transformations such as rotation, reflection and translation, and the scalar value is obtained…
We present group equivariant capsule networks, a framework to introduce guaranteed equivariance and invariance properties to the capsule network idea. Our work can be divided into two contributions. First, we present a generic routing by agreement algorithm defined on elements of a group and prove that equivariance of output pose vectors, as well as invariance of output activations, hold under certain conditions. Second, we connect the resulting equivariant capsule networks with work from the field of group convolutional networks. Through this connection, we provide intuitions of how both meth…
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 …
Qualcomm AI Research（高通人工智能研究院）；University of Amsterdam（阿姆斯特丹大学）
The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep…
Chalmers University of Technology（查尔姆斯理工大学）；Gothenburg University（哥德堡大学）；Umeå University（于默奥大学）；Zenseact
We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\mathcal{M}$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces $\mathcal{M}=G/K$, which are instead equivariant with respect to the global symmetry $G$ on $\mathcal{M}$. Group equivariant layers can be interpreted as intertwiners between ind…
University of Southern California（南加州大学）；USC Institute for Creative Technologies（南加州大学创意技术研究所）；Tencent Game AI Research Center（腾讯游戏AI研究中心）；Pinscreen
Features that are equivariant to a larger group of symmetries have been shown to be more discriminative and powerful in recent studies. However, higher-order equivariant features often come with an exponentially-growing computational cost. Furthermore, it remains relatively less explored how rotation-equivariant features can be leveraged to tackle 3D shape alignment tasks. While many past approaches have been based on either non-equivariant or invariant descriptors to align 3D shapes, we argue that such tasks may benefit greatly from an equivariant framework. In this paper, we propose an effec…
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}
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…
Transformation-robustness is an important feature for machine learning models that perform image classification. Many methods aim to bestow this property to models by the use of data augmentation strategies, while more formal guarantees are obtained via the use of equivariant models. We recognise that compositional, or part-whole structure is also an important aspect of images that has to be considered for building transformation-robust models. Thus, we propose a capsule network model that is, at once, equivariant and compositionality-aware. Equivariance of our capsule network model comes from…
University of Chinese Academy of Sciences（中国科学院大学）；CAS Center for Excellence in Brain Science and Intelligence Technology（中国科学院脑科学与智能技术卓越创新中心）
Weakly supervised semantic segmentation has attracted much research interest in recent years considering its advantage of low labeling cost. Most of the advanced algorithms follow the design principle that expands and constrains the seed regions from class activation maps (CAM). As well-known, conventional CAM tends to be incomplete or over-activated due to weak supervision. Fortunately, we find that semantic segmentation has a characteristic of spatial transformation equivariance, which can form a few self-supervisions to help weakly supervised learning. This work mainly explores the advantag…
3D object detection received increasing attention in autonomous driving recently. Objects in 3D scenes are distributed with diverse orientations. Ordinary detectors do not explicitly model the variations of rotation and reflection transformations. Consequently, large networks and extensive data augmentation are required for robust detection. Recent equivariant networks explicitly model the transformation variations by applying shared networks on multiple transformed point clouds, showing great potential in object geometry modeling. However, it is difficult to apply such networks to 3D object d…
Objects in aerial images have greater variations in scale and orientation than in typical images, so detection is more difficult. Convolutional neural networks use a variety of frequency- and orientation-specific kernels to identify objects subject to different transformations; these require many parameters. Sampling equivariant networks can adjust sampling from input feature maps according to the transformation of the object, allowing a kernel to extract features of an object under different transformations. Doing so requires fewer parameters, and makes the network more suitable for represent…
We introduce a unified framework for group equivariant networks on homogeneous spaces derived from a Fourier perspective. We consider tensor-valued feature fields, before and after a convolutional layer. We present a unified derivation of kernels via the Fourier domain by leveraging the sparsity of Fourier coefficients of the lifted feature fields. The sparsity emerges when the stabilizer subgroup of the homogeneous space is a compact Lie group. We further introduce a nonlinear activation, via an elementwise nonlinearity on the regular representation after lifting and projecting back to the fi…
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 …
This work seeks to improve the generalization and robustness of existing neural networks for 3D point clouds by inducing group equivariance under general group transformations. The main challenge when designing equivariant models for point clouds is how to trade-off the performance of the model and the complexity. Existing equivariant models are either too complicate to implement or very high complexity. The main aim of this study is to build a general procedure to introduce group equivariant property to SOTA models for 3D point clouds. The group equivariant models built form our procedure are…
We propose a method for 3D shape reconstruction from unoriented point clouds. Our method consists of a novel SE(3)-equivariant coordinate-based network (TF-ONet), that parametrizes the occupancy field of the shape and respects the inherent symmetries of the problem. In contrast to previous shape reconstruction methods that align the input to a regular grid, we operate directly on the irregular point cloud. Our architecture leverages equivariant attention layers that operate on local tokens. This mechanism enables local shape modelling, a crucial property for scalability to large scenes. Given …
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…
Stanford University（斯坦福大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Lab（劳伦斯伯克利国家实验室）；Google（谷歌）
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, physi…
Chalmers University of Technology（查尔姆斯理工大学）；Berlin Institute of Technology（柏林工业大学）；Berlin Institute for the Foundations of Learning and Data (BIFOLD)（柏林学习与数据基础研究所（BIFOLD））；University of Gothenburg（哥德堡大学）；Umeå University（于默奥大学）；Zenseact（Zenseact（智己汽车科技有限公司））
We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we f…
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…
Technical University of Munich（慕尼黑工业大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）；Friedrich-Alexander University Erlangen-Nuremberg（埃尔朗根-纽伦堡弗里德里希-亚历山大大学）
Khoury College of Computer Science（胡里计算机学院）；Northeastern University（东北大学）
Transporter Net is a recently proposed framework for pick and place that is able to learn good manipulation policies from a very few expert demonstrations. A key reason why Transporter Net is so sample efficient is that the model incorporates rotational equivariance into the pick module, i.e. the model immediately generalizes learned pick knowledge to objects presented in different orientations. This paper proposes a novel version of Transporter Net that is equivariant to both pick and place orientation. As a result, our model immediately generalizes place knowledge to different place orientat…
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…
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…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Purdue University（普渡大学）
We propose learnable polyphase sampling (LPS), a pair of learnable down/upsampling layers that enable truly shift-invariant and equivariant convolutional networks. LPS can be trained end-to-end from data and generalizes existing handcrafted downsampling layers. It is widely applicable as it can be integrated into any convolutional network by replacing down/upsampling layers. We evaluate LPS on image classification and semantic segmentation. Experiments show that LPS is on-par with or outperforms existing methods in both performance and shift consistency. For the first time, we achieve true shi…
