Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
Classifies Cp-surfaces using equivariant surgery methods.
problem Classifying closed, connected 2-manifolds with cyclic group actions.
method Equivariant surgery methods
result Provides representatives of each isomorphism class using surgery operations.
This paper classifies equivariant principal bundles over a 2-sphere using isotropy representations.
problem Classifying equivariant principal bundles over the 2-sphere.
method Using isotropy representations to classify bundles over the 2-sphere.
result Equivariant principal bundles over the 2-sphere can be classified by a Γ-fixed set of homotopy classes of maps and first Chern class.
Classifies 4D Alexandrov spaces with torus actions.
problem Classifying Alexandrov spaces with torus actions.
method Equivariant classification and homeomorphism analysis.
result Alexandrov spaces are homeomorphic to Riemannian orbifolds.
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.
Defines equivariant holonomy for U(1)-bundles, generalizing properties.
problem Classifying equivariant U(1)-bundles using holonomy.
method Defining equivariant holonomy and showing its classification properties.
result Equivariant U(1)-bundles classified by equivariant holonomy.
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.
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.
Classifies symplectic torus actions up to equivariant symplectomorphism.
problem Classifying symplectic torus actions up to equivariant symplectomorphism.
method Classification theorems based on Duistermaat and Pelayo's work on symplectic torus actions with coisotropic orbits.
result Every almost isotropy-maximal symplectic torus action is equivariantly diffeomorphic to a product of a symplectic toric manifold and a torus.
New rigidity results for complex and quaternionic moment-angle manifolds.
problem Equivariant topological rigidity of complex and quaternionic moment-angle manifolds.
method Reduction to equivariant rigidity of quasitoric (or quoric) quotients and principal bundles.
result Full equivariant rigidity for manifolds with four-dimensional quoric quotients and primary rigidity for higher dimensions.
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.
Optimal classification requires choosing the right group symmetries, contrary to intuition.
problem Improving binary classification performance by selecting appropriate group symmetries.
method Developed a theoretical framework for designing group equivariant neural networks.
result Optimal classification performance is achieved by selecting the appropriate subgroups of symmetries, not the largest equivariant groups.
Constructs 2-vector bundles and 2K-theory for Lie groupoids and 2-equivariant settings.
problem Developing a theory of 2-vector bundles and 2K-theory for Lie groupoids and their equivariant versions.
method Defines 2-vector bundles over Lie groupoids, constructs 2K-theory as Grothendieck completion, and proves classification theorems.
result Establishes an equivalence between homotopy categories of 2-vector bundles and simplicial maps, and computes 2-equivariant 2K-theories for specific Lie groups.
2-knots with S4 symmetry are classified up to equivariant concordance.
problem Classifying 2-knots with S4 symmetry up to equivariant concordance. method Constructing a new invariant called periodic, based on the Arf invariant.
result The smooth equivariant concordance group of 2-knots in S4 is isomorphic to Z/2Z for all d≥2. 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.
New equivariant filters improve graph classification.
problem Designing deep learning models for graph symmetries.
method Nonlinear spectral filters (NLSFs) that are equivariant to graph functional shifts.
result NLSFs outperform existing spectral GNNs in graph classification.
Automatically learns flexible symmetry constraints in neural networks using gradients.
problem Fixed hard constraints on neural network functions that cannot be adapted.
method Improves parameterisations of soft equivariance and optimizes marginal likelihood using differentiable Laplace approximations.
result Achieves equivalent or improved performance on image classification tasks compared to baselines with hard-coded symmetry.
Classifies knots that bound equivariant surfaces with free symmetries.
problem Classifying knots that bound equivariant surfaces with free symmetries.
method Homology cobordism classification of lens spaces using d-invariants.
result Numerical condition determining free periods for torus knots.
Let G be a compact Lie group acting on a smooth manifold M. In this paper, we consider Meinrenken's G-equivariant bundle gerbe connections on M as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to M, and isomorphism cl…
This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the T…
The study classifies equivariant biharmonic maps and proves stability results for certain maps.
problem Classifying and analyzing equivariant biharmonic maps and their stability.
method Generalized biharmonic equation for equivariant maps, improved second variation formula for biharmonic maps.
result No stable proper biharmonic maps with constant square norm of tension field exist from a compact Riemannian manifold into a space form of positive sectional curvature.
Classifies Tk-manifolds with continuous sections.
problem Classifying Tk-manifolds with specific properties. method Uses continuous sections to orbit map for classification.
result Classifies Tk-manifolds up to equivariant diffeomorphism. 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.
We present a novel approach to the classification of conformally equivariant differential operators on spinors in the case of homogeneous conformal geometry. It is based on the classification of solutions for a vector-valued system of partial differential equations, associated to D-modules for the homogeneo…
A 2-torus manifold is a closed smooth manifold of dimension n with an effective action of a 2-torus group (Z2)n of rank n, and it is said to be locally standard if it is locally isomorphic to a faithful representation of (Z2)n on Rn. This paper studies the equivariant classification of locally standar…
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 …
Paper classifies pseudomanifolds over stratified spaces.
problem Classifying pseudomanifolds over stratified spaces.
method Introducing locally standard T-pseudomanifolds and using characteristic data. result Locally standard T-pseudomanifolds over topological stratified pseudomanifolds are classified by their characteristic data. 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.
Automates galaxy morphology classification with less human labelling.
problem Insufficient human-labeled galaxy images for accurate classification.
method Developed a VAE with equivariant transformer layers and a classifier network.
result Improves accuracy with fewer labels and unlabelled data.
ETs improve model robustness to transformations in images.
problem Improving model robustness to predefined transformations.
method Equivariant Transformers (ETs) incorporating functions equivariant to continuous transformation groups.
result ETs achieve up to 15% relative improvement in error rate on image classification tasks.
We give three sufficient criteria for two quasitoric manifolds (M,M') to be (weakly) equivariantly homeomorphic. We apply these criteria to count the weakly equivariant homeomorphism types of quasitoric manifolds with a given cohomology ring.
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.
Causal classification of three Misner-type spacetimes.
problem Causal structure and isocausality of three spacetimes.
method Formal proof of pairwise isocausality on covers and compactified spacetimes.
result Explicit causal bijections and deck-equivariance criterion for isocausality.
Equivariant trisections for group actions on 4-manifolds are introduced and studied.
problem Understanding the equivariant topology of G-manifolds and their quotients. method Introducing G-equivariant trisections and bridge trisections, and establishing their existence for G-manifolds. result Any G-manifold X admits a G-equivariant trisection such that a G-invariant surface S is in equivariant bridge trisection position. 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…
In this paper, a classification of free involutions on 3-dimensional homotopy complex projective spaces is given. By the Z2-equivariant Montgomery-Yang correspondence, we obtain all smooth involutions on S6 with fixed-point set an embedded S3.
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.
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)…
Study of Lagrangian mean curvature flow with equivariant symmetry.
problem Understanding singularities in Lagrangian mean curvature flow.
method Structural theorems about blowups of finite-time singularities.
result Classification of singularities in equivariant case.
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.
Paper tackles rDR classification and lesion segmentation using self-supervised equivariant learning and attention-based MIL.
problem Classifying rDR and segmenting lesions from image-level labels.
method Integrates self-supervised equivariant attention mechanism (SEAM) with attention-based multi-instance learning (MIL).
result Achieved AU ROC of 0.958 on Eyepacs dataset, outperforming state-of-the-art.
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.
New K-theory approach classifies anyonic topological phases in 2D semimetals.
problem Classifying interacting topological phases remains open.
method TED K-theory of configuration spaces of points in the Brillouin torus.
result Classifies su(2)-anyonic topological order in 2D semimetals.
The equivariant CR minimal immersions from the round 3-sphere S3 into the complex projective space CPn have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant,…
New robustness measure accounts for task-specific symmetries.
problem Traditional robustness measures fail for tasks with inherent symmetries.
method Sound notion of adversarial robustness for equivariant tasks, using randomized smoothing and graph edit distance certificates.
result Provable robustness can be achieved for various tasks with inherent symmetries.
We give a new proof of McShane's classification of simple cuspidal geodesics, using simple equivariant methods in the hyperbolic plane.
Study of symplectomorphisms on ruled surfaces under circle actions.
problem Homotopy type of equivariant symplectomorphisms on rational ruled surfaces.
method Analysis of action on compatible and invariant almost complex structures, use of Delzant's and Karshon's classifications.
result Equivariant symplectomorphisms are homotopy equivalent to tori or their pushout.
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…