Machine learning systems may encounter unexpected problems when the data distribution changes in the deployment environment. A major reason is that certain combinations of domains and labels are not observed during training but appear in the test environment. Although various invariance-based algorithms can be applied, we find that the performance gain is often marginal. To formally analyze this issue, we provide a unique algebraic formulation of the combination shift problem based on the concepts of homomorphism, equivariance, and a refined definition of disentanglement. The algebraic requirements naturally derive a simple yet effective method, referred to as equivariant disentangled transformation (EDT), which augments the data based on the algebraic structures of labels and makes the transformation satisfy the equivariance and disentanglement requirements. Experimental results demonstrate that invariance may be insufficient, and it is important to exploit the equivariance structure in the combination shift problem.
This paper introduces equivariant hamiltonian flows, a method for learning expressive densities that are invariant with respect to a known Lie-algebra of local symmetry transformations while providing an equivariant representation of the data. We provide proof of principle demonstrations of how such flows can be learnt, as well as how the addition of symmetry invariance constraints can improve data efficiency and generalisation. Finally, we make connections to disentangled representation learning and show how this work relates to a recently proposed definition.
Khoury College of Computer Sciences, Northeastern University（东北大学库利计算机学院）；Informatics Institute, University of Amsterdam（阿姆斯特丹大学信息学研究所）
Incorporating symmetries can lead to highly data-efficient and generalizable models by defining equivalence classes of data samples related by transformations. However, characterizing how transformations act on input data is often difficult, limiting the applicability of equivariant models. We propose learning symmetric embedding networks (SENs) that encode an input space (e.g. images), where we do not know the effect of transformations (e.g. rotations), to a feature space that transforms in a known manner under these operations. This network can be trained end-to-end with an equivariant task …
How can intelligent agents solve a diverse set of tasks in a data-efficient manner? The disentangled representation learning approach posits that such an agent would benefit from separating out (disentangling) the underlying structure of the world into disjoint parts of its representation. However, there is no generally agreed-upon definition of disentangling, not least because it is unclear how to formalise the notion of world structure beyond toy datasets with a known ground truth generative process. Here we propose that a principled solution to characterising disentangled representations ca…
We present a simple non-generative approach to deep representation learning that seeks equivariant deep embedding through simple objectives. In contrast to existing equivariant networks, our transformation coding approach does not constrain the choice of the feed-forward layer or the architecture and allows for an unknown group action on the input space. We introduce several such transformation coding objectives for different Lie groups such as the Euclidean, Orthogonal and the Unitary groups. When using product groups, the representation is decomposed and disentangled. We show that the presen…
Sapienza, University of Rome（罗马大学）；Università della Svizzera italiana（瑞士意大利语区大学）
We propose a novel approach to disentangle the generative factors of variation underlying a given set of observations. Our method builds upon the idea that the (unknown) low-dimensional manifold underlying the data space can be explicitly modeled as a product of submanifolds. This definition of disentanglement gives rise to a novel weakly-supervised algorithm for recovering the unknown explanatory factors behind the data. At training time, our algorithm only requires pairs of non i.i.d. data samples whose elements share at least one, possibly multidimensional, generative factor of variation. W…
École Normale Supérieure（巴黎高等师范学院）；University of Oxford（牛津大学）
Learning disentangled representations is a key step towards effectively discovering and modelling the underlying structure of environments. In the natural sciences, physics has found great success by describing the universe in terms of symmetry preserving transformations. Inspired by this formalism, we propose a framework, built upon the theory of group representation, for learning representations of a dynamical environment structured around the transformations that generate its evolution. Experimentally, we learn the structure of explicitly symmetric environments without supervision from obse…
University of Tsukuba（筑波大学）；Research Institute for Computational Science Co. Ltd.（计算科学研究所有限公司）
Graphs are one of the most important data structures for representing pairwise relations between objects. Specifically, a graph embedded in a Euclidean space is essential to solving real problems, such as physical simulations. A crucial requirement for applying graphs in Euclidean spaces to physical simulations is learning and inferring the isometric transformation invariant and equivariant features in a computationally efficient manner. In this paper, we propose a set of transformation invariant and equivariant models based on graph convolutional networks, called IsoGCNs. We demonstrate that …
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…
Futurewei Seattle Cloud Lab（未来伟西雅图云实验室）；University of Central Florida（中佛罗里达大学）；Purdue University（普渡大学）
Transformation Equivariant Representations (TERs) aim to capture the intrinsic visual structures that equivary to various transformations by expanding the notion of {\em translation} equivariance underlying the success of Convolutional Neural Networks (CNNs). For this purpose, we present both deterministic AutoEncoding Transformations (AET) and probabilistic AutoEncoding Variational Transformations (AVT) models to learn visual representations from generic groups of transformations. While the AET is trained by directly decoding the transformations from the learned representations, the AVT is tr…
China Mobile Research（中国移动研究院）；Beijing University of Posts and Telecommunications（北京邮电大学）
Previous contrastive learning methods for sentence representations often focus on insensitive transformations to produce positive pairs, but neglect the role of sensitive transformations that are harmful to semantic representations. Therefore, we propose an Equivariant Self-Contrastive Learning (ESCL) method to make full use of sensitive transformations, which encourages the learned representations to be sensitive to certain types of transformations with an additional equivariant learning task. Meanwhile, in order to improve practicability and generality, ESCL simplifies the implementations of…
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…
TUV SUD Singapore（新加坡TUV南德意志集团）；Nanyang Technological University（南洋理工大学）
Autonomous vehicles (AV) have progressed rapidly with the advancements in computer vision algorithms. The deep convolutional neural network as the main contributor to this advancement has boosted the classification accuracy dramatically. However, the discovery of adversarial examples reveals the generalization gap between dataset and the real world. Furthermore, affine transformations may also confuse computer vision based object detectors. The degradation of the perception system is undesirable for safety critical systems such as autonomous vehicles. In this paper, a deep learning system is p…
Wangxuan Institute of Computer Technology, Peking University（北京大学王选计算机技术研究所）；Futurewei Technologies
Recent advances in Graph Convolutional Neural Networks (GCNNs) have shown their efficiency for non-Euclidean data on graphs, which often require a large amount of labeled data with high cost. It it thus critical to learn graph feature representations in an unsupervised manner in practice. To this end, we propose a novel unsupervised learning of Graph Transformation Equivariant Representations (GraphTER), aiming to capture intrinsic patterns of graph structure under both global and local transformations. Specifically, we allow to sample different groups of nodes from a graph and then transform …
Sensory data are often comprised of independent content and transformation factors. For example, face images may have shapes as content and poses as transformation. To infer separately these factors from given data, various ``disentangling'' models have been proposed. However, many of these are supervised or semi-supervised, either requiring attribute labels that are often unavailable or disallowing for generalization over new contents. In this study, we introduce a novel deep generative model, called group-based variational autoencoders. In this, we assume no explicit labels, but a weaker for…
This study tackles unsupervised subword modeling in the zero-resource scenario, learning frame-level speech representation that is phonetically discriminative and speaker-invariant, using only untranscribed speech for target languages. Frame label acquisition is an essential step in solving this problem. High quality frame labels should be in good consistency with golden transcriptions and robust to speaker variation. We propose to improve frame label acquisition in our previously adopted deep neural network-bottleneck feature (DNN-BNF) architecture by applying the factorized hierarchical vari…
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…
University of California San Diego（加州大学圣地亚哥分校）；Northeastern University（东北大学）
Exploiting symmetry in dynamical systems is a powerful way to improve the generalization of deep learning. The model learns to be invariant to transformation and hence is more robust to distribution shift. Data augmentation and equivariant networks are two major approaches to injecting symmetry into learning. However, their exact role in improving generalization is not well understood. In this work, we derive the generalization bounds for data augmentation and equivariant networks, characterizing their effect on learning in a unified framework. Unlike most prior theories for the i.i.d. setting…
In Reinforcement Learning (RL), Convolutional Neural Networks(CNNs) have been successfully applied as function approximators in Deep Q-Learning algorithms, which seek to learn action-value functions and policies in various environments. However, to date, there has been little work on the learning of symmetry-transformation equivariant representations of the input environment state. In this paper, we propose the use of Equivariant CNNs to train RL agents and study their inductive bias for transformation equivariant Q-value approximation. We demonstrate that equivariant architectures can dramati…
Learning representations through deep generative modeling is a powerful approach for dynamical modeling to discover the most simplified and compressed underlying description of the data, to then use it for other tasks such as prediction. Most learning tasks have intrinsic symmetries, i.e., the input transformations leave the output unchanged, or the output undergoes a similar transformation. The learning process is, however, usually uninformed of these symmetries. Therefore, the learned representations for individually transformed inputs may not be meaningfully related. In this paper, we propo…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；Imperial College London（帝国理工学院）；Qualcomm AI Research（高通人工智能研究院）
Equivariance to symmetries has proven to be a powerful inductive bias in deep learning research. Recent works on mesh processing have concentrated on various kinds of natural symmetries, including translations, rotations, scaling, node permutations, and gauge transformations. To date, no existing architecture is equivariant to all of these transformations. In this paper, we present an attention-based architecture for mesh data that is provably equivariant to all transformations mentioned above. Our pipeline relies on the use of relative tangential features: a simple, effective, equivariance-fr…
McGill University（麦吉尔大学）；Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）
Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, iden…
UvA-Bosch Delta Lab（阿姆斯特丹大学-博世三角洲实验室）；University of Amsterdam（阿姆斯特丹大学）
In this work we seek to bridge the concepts of topographic organization and equivariance in neural networks. To accomplish this, we introduce the Topographic VAE: a novel method for efficiently training deep generative models with topographically organized latent variables. We show that such a model indeed learns to organize its activations according to salient characteristics such as digit class, width, and style on MNIST. Furthermore, through topographic organization over time (i.e. temporal coherence), we demonstrate how predefined latent space transformation operators can be encouraged for…
Brigham and Women’s Hospital（布莱根妇女医院）；Harvard Medical School（哈佛医学院）；CSAIL, Massachusetts Institute of Technology（麻省理工学院计算机科学与人工智能实验室）；Boston Children’s Hospital（波士顿儿童医院）
We demonstrate an object tracking method for 3D images with fixed computational cost and state-of-the-art performance. Previous methods predicted transformation parameters from convolutional layers. We instead propose an architecture that does not include either flattening of convolutional features or fully connected layers, but instead relies on equivariant filters to preserve transformations between inputs and outputs (e.g. rot./trans. of inputs rotate/translate outputs). The transformation is then derived in closed form from the outputs of the filters. This method is useful for applications…
Massachusetts Institute of Technology（麻省理工学院）；Scuola Normale Superiore（比萨高等师范学校）；University of Bologna（博洛尼亚大学）
Group convolutions and cross-correlations, which are equivariant to the actions of group elements, are commonly used in mathematics to analyze or take advantage of symmetries inherent in a given problem setting. Here, we provide efficient quantum algorithms for performing linear group convolutions and cross-correlations on data stored as quantum states. Runtimes for our algorithms are logarithmic in the dimension of the group thus offering an exponential speedup compared to classical algorithms when input data is provided as a quantum state and linear operations are well conditioned. Motivated…
Extensive work has demonstrated that equivariant neural networks can significantly improve sample efficiency and generalization by enforcing an inductive bias in the network architecture. These applications typically assume that the domain symmetry is fully described by explicit transformations of the model inputs and outputs. However, many real-life applications contain only latent or partial symmetries which cannot be easily described by simple transformations of the input. In these cases, it is necessary to learn symmetry in the environment instead of imposing it mathematically on the netwo…
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…
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…
We propose a novel unsupervised generative model that learns to disentangle object identity from other low-level aspects in class-imbalanced data. We first investigate the issues surrounding the assumptions about uniformity made by InfoGAN, and demonstrate its ineffectiveness to properly disentangle object identity in imbalanced data. Our key idea is to make the discovery of the discrete latent factor of variation invariant to identity-preserving transformations in real images, and use that as a signal to learn the appropriate latent distribution representing object identity. Experiments on bo…
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…
Representations learnt through deep neural networks tend to be highly informative, but opaque in terms of what information they learn to encode. We introduce an approach to probabilistic modelling that learns to represent data with two separate deep representations: an invariant representation that encodes the information of the class from which the data belongs, and an equivariant representation that encodes the symmetry transformation defining the particular data point within the class manifold (equivariant in the sense that the representation varies naturally with symmetry transformations).…
The goal of a classification model is to assign the correct labels to data. In most cases, this data is not fully described by the given set of labels. Often a rich set of meaningful concepts exist in the domain that can much more precisely describe each datapoint. Such concepts can also be highly useful for interpreting the model's classifications. In this paper we propose a model, denoted as Variational Autoencoder-based Contrastive Explanation (VAE-CE), that represents data with high-level concepts and uses this representation for both classification and generating explanations. The explana…
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…
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…
Equivariant-Aware Structured Pruning for Efficient Edge Deployment: A Comprehensive Framework with Adaptive Fine-Tuning
Faculty of Computers and Information Technology, University of Tabuk（计算机与信息科技学院，塔布克大学）；Artificial Intelligence and Sensing Technologies (AIST) Research Center, University of Tabuk（人工智能与传感技术（AIST）研究中心，塔布克大学）
University of British Columbia（不列颠哥伦比亚大学）；McGill University（麦吉尔大学）；Mila（米拉研究所）
The relational model is a ubiquitous representation of big-data, in part due to its extensive use in databases. In this paper, we propose the Equivariant Entity-Relationship Network (EERN), which is a Multilayer Perceptron equivariant to the symmetry transformations of the Entity-Relationship model. To this end, we identify the most expressive family of linear maps that are exactly equivariant to entity relationship symmetries, and further show that they subsume recently introduced equivariant maps for sets, exchangeable tensors, and graphs. The proposed feed-forward layer has linear complexit…
MIT（麻省理工学院）；Facebook AI Research（脸书人工智能研究院）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
In state-of-the-art self-supervised learning (SSL) pre-training produces semantically good representations by encouraging them to be invariant under meaningful transformations prescribed from human knowledge. In fact, the property of invariance is a trivial instance of a broader class called equivariance, which can be intuitively understood as the property that representations transform according to the way the inputs transform. Here, we show that rather than using only invariance, pre-training that encourages non-trivial equivariance to some transformations, while maintaining invariance to ot…
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 Condition…
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…
South China Normal University（华南师范大学）；Research Institute of China Telecom Corporate Ltd.（中国电信股份有限公司研究院）
Introducing prior auxiliary information from the knowledge graph (KG) to assist the user-item graph can improve the comprehensive performance of the recommender system. Many recent studies show that the ensemble properties of hyperbolic spaces fit the scale-free and hierarchical characteristics exhibited in the above two types of graphs well. However, existing hyperbolic methods ignore the consideration of equivariance, thus they cannot generalize symmetric features under given transformations, which seriously limits the capability of the model. Moreover, they cannot balance preserving the het…
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…
