Unified method for CNNs to approximate equivariant maps across various groups.
problem Limited universal approximation theorems for CNNs with specific groups and settings.
method Unified approach to derive universal approximation theorems for equivariant maps by CNNs in diverse settings.
result Ability to handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.
L-CNNs preserve gauge symmetry in neural networks.
problem Applying machine learning to lattice gauge theory while preserving gauge symmetry.
method L-CNNs use gauge equivariance to construct a gauge equivariant convolutional layer and bilinear layer.
result L-CNNs achieve higher accuracy in non-linear regression tasks compared to non-equivariant CNNs.
L-CNNs maintain gauge symmetry on non-Abelian lattice theories.
problem Applying convolutional neural networks to non-Abelian lattice gauge theories while preserving gauge symmetry.
method Developed a geometric formulation of L-CNNs that are equivariant under global symmetries and gauge transformations.
result Convolutional operations in L-CNNs are a specific case of gauge-equivariant neural networks on SU(N) principal bundles. Group-equivariant subsampling layers improve CNNs' equivariance.
problem Non-translation equivariance in subsampling operations.
method Translation and group-equivariant subsampling/upsampling layers.
result Group-equivariant autoencoders learn equivariant representations.
L-CNNs preserve gauge symmetry in lattice simulations.
problem Breaking gauge symmetry in neural network models.
method Lattice gauge equivariant convolutional neural networks (L-CNNs).
result L-CNNs represent gauge invariant functions on the lattice.
Equivariant CNNs improve RL performance in symmetric environments.
problem Learning equivariant representations for RL in symmetric environments.
method Proposed and studied equivariant CNNs for RL.
result Equivariant CNNs enhance RL performance and sample efficiency.
Gauge equivariant CNNs improve image segmentation and climate pattern analysis.
problem Improving image segmentation and climate pattern analysis on non-Euclidean surfaces.
method Developed gauge equivariant convolutional neural networks (gCNNs) for icosahedral surfaces.
result Significant improvements in image segmentation and climate pattern analysis.
Scale-equivariant CNNs handle scale changes for improved performance.
problem Translation equivariance is not sufficient for handling scale changes in CNNs.
method Developed scale-equivariant convolutional networks with steerable filters.
result Demonstrated state-of-the-art results on MNIST-scale and STL-10 datasets.
Graph-based CNN for spherical data with equivariance.
problem Efficiently learning from non-uniformly distributed spherical data.
method Discretized sphere as graph, graph convolutions, equivariance using Defferrard's graph neural network.
result Good performance on rotation-invariant learning problems.
ScDCFNet improves multiscale image classification with reduced model size.
problem Improving performance in multiscale image classification.
method Decomposed convolutional filters for ST-equivariant CNNs.
result ScDCFNet achieves significantly improved performance in multiscale image classification.
We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic cla…
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 s…
L-CNNs learn gauge invariant quantities on lattices.
problem Learning gauge invariant quantities on lattices.
method Novel convolutional layer preserving gauge equivariance and forming Wilson loops.
result L-CNNs can approximate any gauge covariant function on the lattice.
PDE-based G-CNNs add geometric symmetries to CNNs without augmentation.
problem Designing CNNs with built-in symmetries like rotation.
method Formulate CNN layers as PDE solvers on homogeneous spaces.
result PDE-G-CNNs achieve better performance with fewer parameters.
Proposes a new CNN for meshes that can handle orientation.
problem Isotropic kernels in graph convolutions are insensitive to mesh geometry.
method Introduces gauge equivariant kernels and geometric message passing.
result Significantly improved expressivity over conventional GCNs.
B-Spline CNNs on Lie Groups expand G-CNNs to arbitrary groups.
problem Leveraging geometric structure for improved feature learning.
method Lifting feature maps to B-spline expansions on Lie algebra.
result G-CNNs on Lie groups outperform classical 2D CNNs.
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 …
DeepSphere improves spherical CNNs by balancing efficiency and rotation equivariance.
problem Designing efficient and rotation-equivariant convolutional layers for spherical data.
method Graph-based approach to represent spherical data, focusing on the number of vertices and neighbors.
result DeepSphere achieves state-of-the-art performance and demonstrates efficiency and flexibility.
Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typic…
Geometric models improve feature extraction and equivariance in image generation.
problem Improving feature extraction at multiscale levels and reducing network complexity.
method Proposes a geometric generative model based on morphological PDEs and GANs, incorporating equivariance for geometric interpretability.
result Preliminary results show GM-GAN outperforms classical GANs on MNIST data.
New CNNs for set functions using powerset shifts.
problem Classifying data indexed by the powerset of a finite set.
method Developed a novel class of CNNs with shift-equivariant convolutions.
result Demonstrated potential of powerset CNNs on synthetic and real-world datasets.
Wavelet Networks learn from raw time-series data, outperforming conventional CNNs.
problem Learning from raw time-series data efficiently and effectively.
method Constructing scale-translation equivariant neural networks based on wavelet symmetries.
result Wavelet Networks outperform conventional CNNs on raw waveforms and spectrograms.
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 …
GNNs generalize CNNs for graph data, showing equivariance and stability.
problem Processing signals on graphs.
method Graph convolutional filters, nonlinearities, stacked layers.
result GNNs converge to graphon neural networks under graph convergence.
Covariance in physics and CNNs share similarities, with simple assumptions uniquely determining convolution forms.
problem Understanding the similarities between physics and CNNs using covariance.
method Examined similarities and differences, and demonstrated that simple assumptions lead to unique convolution forms.
result Simple assumptions of covariance, locality, linearity, and weight sharing uniquely determine convolution forms.
We introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular 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 analysi…
Measures equivariance in vision models using Lie derivative.
problem Understanding the role of equivariance in recent vision models.
method Introducing Lie derivative to measure equivariance with strong mathematical foundations and minimal hyperparameters.
result Many violations of equivariance can be linked to spatial aliasing in network layers, and larger models tend to display more equivariance.
The paper generalizes equivariant neural networks on homogeneous spaces to the non-linear setting.
problem Equivariant neural networks on homogeneous spaces.
method Deriving generalized steerability constraints for non-linear equivariant layers.
result The universality of the derived construction for non-linear equivariant layers.
Paper introduces method to make neural networks symmetrical.
problem Creating symmetrical neural networks for data with inherent symmetries.
method Introduces a method for modifying neural networks to enforce equivariance.
result Group convolutional neural networks are a special case of the introduced framework.
It has long been recognized that the invariance and equivariance properties of a representation are critically important for success in many vision tasks. In this paper we present Steerable Convolutional Neural Networks, an efficient and flexible class of equivariant convolutional networks. We show that steerable CNNs …
L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.
problem Approximating gauge actions with lattice artifacts.
method Lattice gauge-equivariant convolutional neural networks (L-CNNs).
result L-CNNs provide fixed point actions with no lattice artifacts.
Group equivariant neural networks simplify complex tasks with group representation theory.
problem Challenging tasks requiring input transformations like rotations.
method Group representation theory, non-commutative harmonic analysis, differential geometry.
result A neural network is group equivariant if and only if it has a convolutional structure.
New framework uses symmetry-based matrices for efficient, flexible NNs.
problem Designing neural networks with relaxed equivariance.
method Symmetry-based structured matrices, Group Matrices (GMs).
result GMs enable competitive performance with fewer parameters.
Coordinate-independent convolutions on manifolds avoid reference frame ambiguity.
problem Applying convolutions on non-Euclidean manifolds without reference frame ambiguity.
method Developed coordinate-independent and gauge-equivariant convolutions on Riemannian manifolds.
result Coordinate-independent convolutions are equivariant under local gauge transformations.
Universal MLPs with a single hidden layer can learn any function.
problem Learning on various data structures like sequences, images, sets, and graphs.
method Using group theory, the paper proves the universality of a broad class of equivariant MLPs with a single hidden layer.
result Having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance).
Generalizes CNNs for Lie group equivariance across various data types.
problem Equivariance to transformations like rotations for non-image data.
method Constructs equivariant convolutional layers for Lie groups.
result Models conserve linear and angular momentum in Hamiltonian systems.
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 introdu…
Study on geodesic distances on SE(3)/SO(2) in machine learning.
problem Investigating the efficiency of computationally efficient sections in selecting geodesic distances.
method Analyzing geodesic distances on reductive homogeneous spaces, proving the efficiency of minimal distance sections.
result Minimal distance sections are not always geodesic minimizers, but minimal horizontal geodesics are.
Equivariant neural network simplifies particle physics models.
problem Complexity and interpretability in particle physics classification.
method Lorentz group equivariant neural network architecture.
result Simplified, interpretable models with fewer parameters.
This work relaxes GNN symmetries to approximate automorphisms, improving model performance.
problem Improving graph neural network performance on asymmetric graphs.
method Formalizing approximate symmetries via graph coarsening, introducing a bias-variance formula.
result Best generalization performance achieved by choosing a larger symmetry group than automorphisms but smaller than permutations.
The paper introduces models to learn generalized transformation equivariant representations.
problem Capturing intrinsic visual structures equivariant to various transformations.
method Deterministic and probabilistic AutoEncoding Transformations (AET and AVT) models trained to learn visual representations from generic groups of transformations.
result Generalized TERs (GTERs) that are equivariant to transformations in a more general fashion.
New approximative kernels improve PDE-G-CNNs for geometric deep learning.
problem Inaccurate approximations of exact kernels in PDE-G-CNNs.
method Developed new approximative kernels that work regardless of spatial anisotropy.
result New kernels provide better error estimates and maintain reflectional symmetries.
The paper introduces MDP homomorphic networks for faster reinforcement learning.
problem Current reinforcement learning approaches do not exploit symmetries in the joint state-action space.
method Equivariant neural networks with group-structured symmetries (reflections, rotations).
result MDP homomorphic networks converge faster than unstructured baselines on various tasks.
Classical models describe primary visual cortex (V1) as a filter bank of orientation-selective linear-nonlinear (LN) or energy models, but these models fail to predict neural responses to natural stimuli accurately. Recent work shows that models based on convolutional neural networks (CNNs) lead to much more accurate p…
Convolutional Neural Networks (CNNs) have become the method of choice for learning problems involving 2D planar images. However, a number of problems of recent interest have created a demand for models that can analyze spherical images. Examples include omnidirectional vision for drones, robots, and autonomous cars, mo…
In this paper, we propose Generative Adversarial Network (GAN) architectures that use Capsule Networks for image-synthesis. Based on the principal of positional-equivariance of features, Capsule Network's ability to encode spatial relationships between the features of the image helps it become a more powerful critic in…
Performance of neural networks can be significantly improved by encoding known invariance for particular tasks. Many image classification tasks, such as those related to cellular imaging, exhibit invariance to rotation. We present a novel scheme using the magnitude response of the 2D-discrete-Fourier transform (2D-DFT)…