Equivariant neural networks use symmetry to interpret complex data.
problem Interpreting and understanding the behavior of equivariant neural networks.
method Decompose layers into simple representations and analyze nonlinear activation functions.
result Equivariant neural networks can be interpreted using a filtration generalizing Fourier series.
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.
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.
Characterizes a specific type of neural network for alternating group equivariance.
problem Understanding and characterizing neural networks with alternating group equivariance.
method Characterization of all possible An-equivariant neural networks using tensor powers of Rn. result Found a basis of matrices for learnable, linear An-equivariant layer functions. New fusion blocks improve equivariant neural networks for molecular dynamics.
problem Designing equivariant neural networks for tasks with global symmetries.
method Using fusion diagrams from tensor networks to design novel equivariant components.
result Improved performance with fewer parameters on chemical problems.
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.
New model learns graph neural networks equivariant to various transformations.
problem Learning equivariant graph neural networks for complex transformations.
method E(n)-Equivariant Graph Neural Networks (EGNNs) that are computationally efficient and scalable.
result Achieves competitive or better performance without higher-order representations.
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…
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. Equivariant neural networks improve performance and generalization in complex scalar field theory tasks.
problem Improving performance and generalization in neural networks for complex scalar field theory tasks.
method Incorporating translational equivariance into neural network architectures.
result Equivariant neural networks significantly outperform non-equivariant networks in various tasks, including those beyond the training set and across different lattice sizes.
New neural networks for non-commutative data.
problem No existing neural networks suitable for non-commutative data.
method Developed compact matrix quantum group equivariant neural networks.
result Characterized weight matrices for easy compact matrix quantum groups.
New method breaks symmetry in neural networks, improving sample efficiency.
problem Symmetry in neural networks limits their ability to learn unique features.
method Introduces 'relaxed equivariance' to overcome symmetry limitations.
result Equivariant multilayer perceptrons (E-MLPs) can now break symmetry at the sample level.
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. Rota…
Translationally equivariant neural networks improve performance and generalization in physics problems.
problem Performance and generalization issues in machine learning applied to physics problems.
method Investigation of translationally equivariant convolutional neural networks for complex scalar field theory on a 2D lattice.
result Translationally equivariant neural networks significantly outperform non-equivariant architectures in various regression and classification tasks.
Equivariant neural networks improve performance and generalization in lattice field theory tasks.
problem Improving neural network performance and generalization in lattice field theory.
method Investigation of translationally equivariant neural networks in a two-dimensional scalar field model.
result Equivariant neural networks significantly outperform non-equivariant ones in various tasks, including physical parameters and lattice sizes.
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.
New neural network processes 3D volumes with improved equivariance.
problem Improving neural network performance on 3D volumes with symmetries.
method Equivariant neural network using moving frames approach.
result Trained model outperforms benchmarks in medical volume classification.
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.
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.
New method finds Lie group representations without explicit groups, enabling new neural network architectures.
problem Building neural networks equivariant to arbitrary Lie groups.
method Algorithm to find Lie group representations from Lie algebra structure constants. Self-contained method for constructing Lie group-equivariant neural networks.
result First object-tracking model equivariant to the Poincaré group.
Characterizes group-equivariant neural networks for three groups.
problem Understanding equivariant neural networks for orthogonal, special orthogonal, and symplectic groups.
method Characterized all possible group-equivariant neural networks for three groups.
result Found spanning sets of matrices for learnable, linear equivariant layer functions.
SymPE breaks symmetries in equivariant networks, improving performance across various tasks.
problem Equivariant networks cannot break symmetries, leading to poor performance in tasks with symmetrical inputs.
method Novel equivariant conditional distributions and randomized canonicalization.
result SymPE significantly improves performance of group-equivariant and graph neural networks.
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.
Proposes SE(3) equivariant graph neural networks with local frames for efficient geometric approximation.
problem Equivariance in deep learning for arbitrary transformations, especially in physics.
method Introduces SE(3) equivariant graph neural networks with complete local frames to efficiently approximate geometric quantities.
result Achieves best or competitive performance in Newton mechanics modeling and equilibrium molecule conformation generation.
Novel CG-EGNNs learn equivariant functions from Clifford algebras.
problem Lack of equivariance in high-order graph neural networks.
method Integrates high-order local structures with Clifford algebras for equivariant learning.
result CG-EGNNs outperform previous methods on various benchmarks.
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.
In this paper, we develop a theory about the relationship between G-invariant/equivariant functions and deep neural networks for finite group G. Especially, for a given G-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip G-actions and each affine t…
New method connects neural networks to diagrammatic algebra.
problem Constructing permutation equivariant neural networks.
method Schur-Weyl duality between symmetric group and partition algebra.
result Simple diagrammatic method for calculating weight matrices.
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.
Gradient descent struggles to learn equivariant neural networks, even with symmetries.
problem Learning equivariant neural networks via gradient descent is hard.
method Lower bounds for various equivariant neural network classes.
result Gradient descent struggles to learn equivariant neural networks, even with symmetries.
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.
This paper explores how enforcing equivariance constraints limits neural network expressivity and proposes compensatory model size increases.
problem The impact of enforcing equivariance constraints on the expressive power of neural networks.
method Examined 2-layer ReLU networks, analyzed boundary hyperplanes and channel vectors, and constructed upper bounds on model size required for compensation.
result Enforcing equivariance constraints reduces the expressive power of neural networks, but this can be compensated by increasing model size.
Category theory enhances understanding of group-equivariant neural networks.
problem Understanding and working with group-equivariant neural networks.
method Application of category theory to tensor power spaces of Rn for groups Sn, O(n), Sp(n), and SO(n). result New insights and an algorithm for computing equivariant linear layers.
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.
Paper benchmarks quantum neural networks against classical ones for binary classification tasks.
problem Comparing quantum neural networks with classical ones for binary classification.
method Evaluated with two toy examples, focusing on model complexity and training data size.
result EQNN and QNN outperform ENN and DNN for smaller parameter sets and training data samples.
New neural networks respect symmetries in symmetric tensors, improving efficiency and generalization.
problem Learning from symmetric tensors efficiently and respecting their inherent symmetries.
method Developed two characterizations of linear permutation equivariant functions between symmetric power spaces of R^n.
result These functions are highly data efficient compared to standard MLPs and generalize well to different sizes of symmetric tensors.
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 architect…
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.
Graph neural networks improve systemic risk measures for financial networks.
problem Computing systemic risk measures for graph-structured financial networks.
method Extended permutation equivariant neural networks (X-PENNs) for numerical approximation.
result Graph neural networks outperform other methods in approximating optimal allocations.
Natural graph networks are a new class of graph neural networks that are more flexible and scalable.
problem Traditional graph neural networks are limited by equivariance to node permutations.
method Introduced natural graph networks, which are more flexible and scalable than conventional graph neural networks.
result Natural graph networks are as scalable as conventional message passing graph neural networks but more flexible.
Develops a theory for equivariant networks with partial domain symmetry.
problem Limited analysis of equivariant networks with partial domain symmetry.
method Proposes pointwise definitions of correct, incorrect, and extrinsic equivariance.
result Establishes error lower bounds for networks with partial symmetry.
Study improves generalization bounds for equivariant networks on Markov data.
problem Challenges in integrating equivariance with Markov dependencies in neural networks.
method Applied McDiarmid's inequality and computed covering number using group theory.
result Derived upper bound on Rademacher complexity for equivariant neural networks on Markov datasets.
Sign equivariant networks improve model expressiveness for spectral geometric learning.
problem Limited expressiveness of sign invariant models for tasks like graph link prediction.
method Developed sign equivariant neural network architectures based on new analytic sign equivariant polynomials.
result Sign equivariant models achieve theoretical benefits in spectral geometric learning tasks.
New method constructs equivariant neural networks for arbitrary matrix groups.
problem Challenges in constructing equivariant neural networks for complex groups.
method Completely general algorithm for solving equivariant layers of matrix groups.
result Constructs multilayer perceptrons equivariant to multiple groups including O(1,3), O(5), Sp(n), and Rubik's cube group.
Frame Averaging makes neural networks invariant or equivariant to new symmetries.
problem Designing neural networks that respect symmetries while being expressive and efficient.
method Introduces Frame Averaging (FA) as a systematic framework to adapt architectures to become invariant or equivariant to new symmetries.
result Frame Averaging guarantees exact invariance or equivariance while being simpler to compute than full group averaging.
Introduces P-tensors for generalized higher-order message passing in graph neural networks.
problem Expanding the expressive power of graph neural networks through higher-order structures.
method Introduces P-tensors to define the most general form of permutation equivariant message passing.
result Achieves state-of-the-art performance on molecular datasets.
Paper presents a novel neural network for MIMO symbol detection.
problem Handling a variable number of users in MIMO systems.
method Recurrent and permutation equivariant neural network architecture with iterative decoding.
result The neural detector outperforms existing methods in accuracy and efficiency.
New neural networks learn graph symmetries.
problem Learning from graph data without considering vertex relations.
method Constructs equivariant neural networks to Aut(G) group.
result Characterizes learnable, linear, Aut(G)-equivariant functions.