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.
An algorithm for efficient computation of equivariant neural network layers.
problem Efficiently computing with Brauer's group equivariant neural network layers.
method Category theoretic constructions and Kronecker product matrices.
result Significant reduction in computational cost compared to naive implementation.
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.
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.
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.
New algorithm speeds up group equivariant neural networks computations.
problem Challenging computations in group equivariant neural networks.
method Diagrammatic framework based on category theory for matrix multiplication.
result Exponential improvement in time complexity for matrix multiplication.
Unified framework for image restoration using equivariant denoisers.
problem Restoring images with realistic priors and invariant transformations.
method Unified framework named ERED based on equivariant denoisers and stochastic optimization.
result Unified framework ERED converges and improves image restoration.
We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic…
Enhances quantum computing for symmetrical systems, proving a new class of problems.
problem Proving the efficiency of a new quantum computing model for symmetrical systems.
method Introducing equivariant convolutional quantum algorithms tailored for SU(d) symmetries.
result Demonstrates a problem that can be solved efficiently on a new quantum model, suggesting it's not classically simulatable.
LEAPS samples discrete distributions via CTMCs and locally equivariant networks.
problem Sampling from discrete distributions with known normalization.
method Continuous-time Markov chain, locally equivariant functions, attention layers, convolutional networks.
result LEAPS minimizes the variance of importance weights, improving sampling efficiency.
EDGI improves sample efficiency and generalization in tasks with spatial and temporal symmetries.
problem Sample inefficiency and poor generalization in tasks with geometric symmetries.
method Equivariant Diffuser framework, SE(3)xZxSn-equivariant diffusion model.
result EDGI is more sample efficient and generalizes better than non-equivariant models.
Spin networks boost quantum algorithms solving SU(2) symmetric problems.
problem Efficiently solving SU(2) symmetric problems on quantum hardware.
method Using SU(2) equivariant variational quantum circuits based on spin networks.
result Spin networks provide a direct implementation for SU(2) equivariant quantum circuits.
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.
The paper defines and calculates an upper bound for the equivariant crossing number of two-bridge knots.
problem Finding the minimum number of crossings in symmetric diagrams for two-bridge knots.
method Defining and calculating c2(K) for two-bridge knots by restricting diagrams to two types. result An algorithm to determine c2(K) for any two-bridge knot and results up to 14 crossings. 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.
We consider groups G which have a cocompact, 3-manifold model for the classifying space \underline{E}G. We provide an algorithm for computing the rationalized equivariant K-homology of \underline{E}G. Under the additional hypothesis that the quotient 3-orbifold \underline{E}G/G is geometrizable, the rationalized K-homo…
Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.
problem Equivariant invariants on manifolds with boundary.
method Analysis of Dirac operators, winding numbers, spectral flow, Maslov indices, and η-invariants.
result Established relation between equivariant η-invariants and Maslov triple indices.
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.
Study of equivariant ribbon concordance using Khovanov homology.
problem Understanding equivariant ribbon concordance.
method Functoriality of equivariant Khovanov homology under equivariant cobordisms, and induced split injection.
result Equivariant ribbon concordances induce a split injection on equivariant Khovanov homology.
Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.
problem Equivariant algebraic concordance of strongly invertible knots.
method Defining a homomorphism Φ from equivariant concordance group to a new equivariant algebraic concordance group, proving it lifts known homomorphisms and provides new obstructions. result Obtains a new obstruction to equivariant sliceness and novel lower bounds on equivariant slice genus.
Infinite G-invariant minimal hypersurfaces found in Riemannian manifolds.
problem Finding minimal hypersurfaces in manifolds with group actions.
method New algorithm using multi-stage maximal cuttings.
result Each G-homology class admits infinitely many distinct realizations by embedded minimal G-hypersurfaces. Defines an equivariant Ruelle dynamical zeta function for flows on manifolds.
problem Defining a zeta function for equivariant flows on manifolds.
method Equivariant generalization of Guillemin's trace formula.
result Computes the equivariant Ruelle zeta function in various examples.
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.
Study of equivariant movie moves for involutive links.
problem Equivariant cobordisms between involutive links.
method Equivariant Morse theory and singularity theory.
result 39 equivariant movie moves for isotopic cobordisms.
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…
A bound on knot unknotting using equivariant signature.
problem Equivariant unknotting of knots.
method Analysis of strongly invertible knots and application of equivariant unknotting moves.
result The equivariant signature provides a lower bound for the equivariant unknotting number.
New proof for 4D symplectic manifolds: equivariant cohomology determines diffeotype.
problem Determining if 4D symplectic manifolds are diffeomorphic based on their equivariant cohomology.
method Proved that equivariant cohomology rings of Hamiltonian circle actions on 4D symplectic manifolds determine their equivariant diffeotypes.
result Isomorphism of equivariant cohomology rings implies equivariant diffeomorphism for 4D symplectic manifolds.
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.
Weisfeiler and Leman enhance graph learning for machine learning tasks.
problem Learning from graph data in machine learning.
method Weisfeiler and Leman algorithm applied to graph and node representation learning.
result The algorithm improves graph and node representation learning in machine learning.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…
New basis for permutation equivariant layers reduces computation costs.
problem Efficiently computing permutation equivariant layers in neural networks.
method Generalized partition algebra basis with low-rank tensors.
result Low-rank tensors enable faster computation compared to orbit basis.
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. 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.
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.
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.
Study on equivariant Heegaard genus of reducible 3-manifolds with group actions.
problem Understanding the equivariant Heegaard genus of reducible 3-manifolds with group actions.
method Thin position theory for 3-dimensional orbifolds to establish bounds on equivariant Heegaard genus.
result Sharp bounds on equivariant Heegaard genus of reducible manifolds, similar to tunnel number results.
We show that J. Lott's equivariant higher analytic torsion for compact group actions depends only on the equivariant Euler characteristic.
New equivariant version of Khovanov homology for annuli.
problem Developing a new mathematical framework for annular Khovanov homology.
method Using Frobenius algebra and equivariant cohomology of CP1. result Introduced an equivariant version of the Temperley-Lieb algebra.
For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We sh…
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.
In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.
We present a 3D capsule module for processing point clouds that is equivariant to 3D rotations and translations, as well as invariant to permutations of the input points. The operator receives a sparse set of local reference frames, computed from an input point cloud and establishes end-to-end transformation equivarian…
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…
Paper develops equivariant basic cohomology for Lie groupoids.
problem Equivariant cohomology for Lie groupoids with weak actions.
method Using Kan fibrations and fiber structures, constructing models and comparing with existing theories.
result Equivariant basic cohomology theory for orbifolds and Lie groupoids.
We construct for an equivariant cohomology theory for proper equivariant CW-complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structur…
In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes character…
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.