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.
Equivariant diffusion model generates 3D molecules efficiently.
problem Generating high-quality 3D molecules efficiently.
method Equivariant Diffusion Model (EDM) that operates on atom coordinates and types.
result Significantly outperforms previous methods in molecule quality and training efficiency.
We define the equivariant holonomy of an invariant connection on a principal U(1)-bundle. The properties of the ordinary holonomy are generalized to the equivariant setting. In particular, equivariant U(1)-bundles with connection are shown to be classified by its equivariant holonomy modulo isomorphisms. We also show t…
Efficiently samples and learns densities with symmetries using equivariant methods.
problem Efficiently sampling and learning densities with symmetries.
method Equivariant Stein Variational Gradient Descent (SVGD) and equivariant energy based models.
result Improves and scales up training of energy based models.
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.
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.
The study examines how equivariance in networks affects generalization error using PAC-Bayesian bounds.
problem Understanding how equivariance in networks impacts generalization error.
method Utilized PAC-Bayesian analysis for equivariant networks, deriving norm-based bounds for generalization error.
result The bound indicates that using larger group size in the model improves generalization error.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both the ordinary smooth Deligne cohomology and the ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant ge…
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.
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.
Improves U-Net for scale equivariance in semantic segmentation.
problem Improving generalization in semantic segmentation tasks with varying scales.
method Introduces Scale Equivariant U-Net (SEU-Net) with carefully applied subsampling and upsampling layers and scale-equivariant layers.
result Significantly improved generalization to different scales compared to U-Net and scale-equivariant architecture without upsampling.
Unified classification of equivariant principal bundles using higher homotopy theory.
problem Unified classification of equivariant principal bundles.
method Smooth Oka principle, singular-cohesive homotopy theory, internally describing principal bundles.
result Unified classification results for equivariant principal bundles.
The paper introduces μK-stability for polarized schemes and develops equivariant calculus.
problem The existence of μ-cscK metrics and their stability. method Develops equivariant calculus and introduces μ-character to study μK-stability. result Derives μ-Futaki invariant and an equivariant first Chern class for general test configurations. 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.
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.
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.
In this paper, we prove an equivariant Kastler-Kalau-Walze type theorem for spin manifolds without boundary. For 6 dimensional spin manifolds with boundary, we also give an equivariant Kastler-Kalau-Walze type theorem. Then we generalize this theorem to the general n dimensional manifold. An equivariant Kastler-Kal…
We investigate an equivariant generalization of Morse theory for a general class of integrable models. In particular, we derive equivariant versions of the classical Poincaré-Hopf and Gauss-Bonnet-Chern theorems and present the corresponding path integral generalizations. Our approach is based on equivariant cohomology…
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
Study symmetries in equivariant Khovanov homology.
problem Understand symmetries in equivariant Khovanov homology.
method Construction of an involution, integral lifting, splitting of theories, and relation to Rasmussen's invariant.
result Established symmetries in equivariant Khovanov homology.
Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.
problem Tackles the limitation of previous equivariant neural networks by considering permutations of matrices.
method Derives formulae for general permutation equivariant layers, including matrix permutations. Presents a second order graph variational encoder.
result Latent distribution of equivariant generative models must be exchangeable.
GSA-Nets apply group equivariance to self-attention for vision tasks.
problem Improving self-attention networks for vision tasks.
method Define group-equivariant positional encodings.
result GSA-Nets outperform non-equivariant self-attention networks on vision benchmarks.
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.
We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must…
Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M. An equivariant Thom form θ on V is a compactly supported closed equivariant form such that its integral along the fib…
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.
SymDiff uses stochastic symmetrisation for equivariant diffusion models.
problem Constructing equivariant diffusion models for data augmentation.
method Stochastic symmetrisation for lightweight, efficient, and easy-to-implement equivariance.
result SymDiff achieves significant empirical benefit for E(3)-equivariant molecular generation. A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic Z/2-equivariant unitary bordism ring, in…
Equivariant MuZero improves generalization in procedurally generated environments.
problem Leveraging world models for robust generalization in real-world scenarios.
method Incorporates symmetries of the environment into MuZero's world-model architecture, making MuZero equivariant to certain groups of transformations.
result Improves data efficiency and generalization capabilities, demonstrated on procedurally generated MiniPacman and Chaser.
Generalizes Floer homotopy via Morse-Bott theory.
problem Constructing equivariant models in Floer theory.
method Morse-Bott theory, flow categories, stable homotopy types.
result Equivalence of Borel equivariant spectra for certain Lagrangians.
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.
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.
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.
Develops approximately equivariant neural processes for better data modeling.
problem Real-world data often breaks exact equivariance; how to model this?
method General approach to creating approximately equivariant architectures, applicable to any model and symmetry group.
result Approximately equivariant neural processes outperform non-equivariant and strictly equivariant models in regression tasks.
New model improves field learning with improved equivariance.
problem Learning equivariant stochastic fields.
method Equivariant Gaussian processes and Steerable Conditional Neural Processes.
result SteerCNPs significantly improve performance in transfer learning tasks.
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.
E-NFs generate molecules and their positions while preserving Euclidean symmetries.
problem Generating molecules with their positions while preserving Euclidean symmetries.
method Integrating E(n) graph neural networks into a differential equation to create an invertible equivariant function.
result E-NFs significantly outperform baselines and existing methods in log-likelihood for particle systems and molecules.
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…
We use contact geometry to describe the monoid of projectively equivariant meromorphic differential operators on a complex curve, quantization of which generalizes known constructions of classical equivariants to non-commutative function algebras in several variables.
We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.
The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "N-F-amenability" or "amenability dimension", and "d-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell--Jones and Bo…
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 dee…
The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.
problem Constructing harmonic maps into symmetric spaces.
method Equivariant primitive harmonic maps construction.
result Examples of S1-equivariant Willmore Moebius strips in S3. Study on equivariant Q-sliceness for strongly invertible knots.
problem Understanding Q-sliceness for strongly invertible knots.
method Constructive and obstructive approaches using Fox-Milnor condition and equivariant concordance.
result Klein amphichiral knots are equivariant Q-slice in a single Q-homology 4-ball.
Paper proves non-zero generalization boost for equivariant models.
problem Improving generalization in machine learning models.
method Analyzes simplest case of linear models, focusing on invariant/equivariant properties.
result First provably non-zero improvement in generalization for invariant/equivariant models.
The study proves a generic multiplicity one theorem for G-invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for G-invariant minimal hypersurfaces. method Equivariant min-max theory and analysis of G-homology classes. result Shows a generic multiplicity one theorem for G-invariant minimal hypersurfaces. In 1993, Bismut and Zhang establish a mod Z embedding formula of Atiyah-Patodi-Singer reduced eta invariants. In this paper, we explain the hidden mod Z term as a spectral flow and extend this embedding formula to the equivariant family case. In this case, the spectral flow is generalized to the equivariant chern chara…