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. 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 …
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 convo…
An impossibility result shows limitations in learning symmetries and equivariant functions.
problem Learning symmetries and equivariant functions simultaneously is impossible under certain conditions.
method Careful study of approximation for groups and semigroups, analysis of neural networks.
result Linearly equivariant networks can be used to learn equivariant functions, but group-convolutional networks have limitations.
The translation equivariance of convolutional layers enables convolutional neural networks to generalize well on image problems. While translation equivariance provides a powerful inductive bias for images, we often additionally desire equivariance to other transformations, such as rotations, especially for non-image d…
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.
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.
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 th…
Lie groupoid equivariant neural networks are a new type of neural network.
problem Designing neural networks that respect the structure of Lie groupoids.
method Introducing Lie groupoid equivariant convolutions and layers, and showing their equivalence to Lie algebroid-equivariant networks.
result Lie groupoid equivariant neural networks are equivalent to certain Lie algebroid-equivariant networks.
This work introduces a method for almost equivariance in neural networks using Lie algebra convolutions.
problem Real-world data often does not conform to strict group equivariances, leading to underperformance in models.
method Definition and practical implementation of almost equivariance through Lie algebra convolutions.
result Demonstrated the validity of the approach through benchmarking against fully equivariant settings.
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.
Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical t…
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.
Rotationally equivariant convolutions improve molecular property prediction.
problem Predicting molecular properties using graph neural networks.
method Ablation study with rotationally equivariant and invariant convolutions on QM9 data set.
result Rotationally equivariant layers decrease test error by an average of 23%.
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…
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 …
GCNNs on homogeneous spaces use vector bundles and Hilbert spaces.
problem Learning data on homogeneous spaces with global symmetry.
method Analysis of G-equivariant convolutional layers on homogeneous G/K spaces, using vector bundles and reproducing kernel Hilbert spaces. result A precise criterion for expressing G-equivariant layers as convolutional layers, leading to stronger results for some groups. 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…
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 t…
IsoGCNs learn invariant and equivariant graph features for efficient simulations.
problem Learning isometric transformation invariant and equivariant features in graphs for simulations.
method Transformation invariant and equivariant Graph Convolutional Networks (IsoGCNs).
result IsoGCNs outperform state-of-the-art methods on geometrical and physical simulation tasks.
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 exte…
We introduce the Convolutional Conditional Neural Process (ConvCNP), a new member of the Neural Process family that models translation equivariance in the data. Translation equivariance is an important inductive bias for many learning problems including time series modelling, spatial data, and images. The model embeds …
ConvNP improves SP prediction with translation equivariance and coherent samples.
problem Predicting stationary stochastic processes with coherent samples.
method Convolutional Neural Processes (ConvNP) with a new maximum-likelihood objective.
result ConvNP outperforms standard NPs and demonstrates strong generalization on various tasks.
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…
Spherical data is found in many applications. By modeling the discretized sphere as a graph, we can accommodate non-uniformly distributed, partial, and changing samplings. Moreover, graph convolutions are computationally more efficient than spherical convolutions. As equivariance is desired to exploit rotational symmet…
New methods improve translation-equivariant neural processes for modeling unknown functions.
problem Modeling unknown latent functions from irregularly sampled measurements.
method Volterra series and set Fourier convolutions to address translation-equivariance and efficiency.
result Improved translation-equivariant neural processes with analytical transparency and linear scalability.
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.
Bayesian convolutional deep sets improve ambiguity in stationary process modeling.
problem Ambiguity in translation equivariant functional representations due to insufficient data points.
method Introduce Bayesian convolutional deep sets with task-dependent stationary prior.
result Improves representation quality compared to kernel smoother and non-parametric models.
Develops a framework for designing quantum neural networks that respect symmetries.
problem Trainability and generalization issues in quantum neural networks.
method Equivariant quantum neural networks (EQNN) for any symmetry group.
result Efficient construction of equivariant layers for EQNNs, including QCNNs.
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.
LieTransformer extends self-attention to Lie groups for improved deep learning tasks.
problem Improving deep learning performance through group equivariant self-attention.
method LieSelfAttention layers that are equivariant to arbitrary Lie groups and their discrete subgroups.
result Competitive experimental results on various tasks.
HKConv learns hyperbolic features by aggregating kernel points.
problem Challenges in learning good hyperbolic representations using Euclidean operations.
method Proposes HKConv, a trainable hyperbolic convolution that correlates local features with kernel points and aggregates them.
result HKConv learns expressive local features according to hyperbolic geometry and enjoys equivariance to permutation and invariance to parallel transport.
We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these net…
Equivariant neural networks are a class of neural networks designed to preserve symmetries inherent in the data. In this paper, we introduce a general method for modifying a neural network to enforce equivariance, a process we refer to as equivarification. We further show that group convolutional neural networks (G-CNN…
ChebLieNet uses Lie groups to create invariant spectral graph networks.
problem Handling anisotropic data in graph neural networks.
method Develops anisotropic convolutional layers on Lie groups with Riemannian metrics.
result Demonstrates the effectiveness of balancing equivariance and invariance.
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)…
Explains equivariant neural networks for machine learning.
problem Understanding equivariance in neural networks.
method Simple mathematical treatment of neural network concepts.
result Clarifies the mathematical basis of equivariant neural networks.
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.
Steerable E(3) Graph Neural Networks incorporate geometric and physical covariant information.
problem Incorporating covariant information like position, force, velocity, or spin in graph neural networks.
method Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that use steerable MLPs to incorporate geometric and physical covariant information.
result SEGNNs improve upon classic linear point convolutions and recent equivariant graph networks that send invariant messages.
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.
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.
In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distribu…
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.
Method learns equivariances from data without custom architecture design.
problem Learning equivariances for tasks without manually designed architectures.
method Reparameterization to learn equivariant parameter sharing.
result Can learn equivariances for any finite group of transformations.
Paper defines mathematical framework for neural network explainability.
problem Neural network explainability and equivariant operators.
method Mathematical framework based on Group Equivariant Non-Expansive Operators (GENEOs) and complexity measures.
result Formal properties and interpretability of Group Equivariant Operators (GEOs) defined.
EquivCNP learns group symmetries for conditional data.
problem Learning conditional models with data symmetries.
method Group equivariant decomposition and Lie group convolutional layers.
result EquivCNP achieves comparable performance and zero-shot generalization.
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.
The study quantifies how many objects can be linearly classified under all views.
problem Understanding the expressivity of group-equivariant representations.
method Generalization of Cover's Function Counting Theorem to quantify separable dichotomies.
result The fraction of separable dichotomies is determined by the fixed space dimension of the group action.