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.
The paper defines new knot genera and finds bounds for stabilization distances.
problem Finding bounds for stabilization distances of symmetric surfaces.
method Defining new knot genera and using them to find bounds.
result Constructs unknotted symmetric 2-spheres without symmetric 3-ball bounds.
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.
Bound on equivariant index for min-max surfaces.
problem Bounding the index of equivariant min-max surfaces.
method Equivariant min-max procedure with group action.
result Equivariant index bound by number of parameters.
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.
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.
Improved upper bound for equivariant Yamabe invariant in 3D.
problem Upper bound for G-equivariant Yamabe invariant in 3-manifolds. method Used topological assumptions to show an upper bound.
result Improved Hebey-Vaugon conjecture in dimension 3.
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.
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.
Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks, have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. Using tools from group theory, this paper proves the universality of a broad class of equivariant M…
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…
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 knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
The paper provides bounds for embedding manifolds into Euclidean spaces with group actions.
problem Finding explicit bounds for embedding manifolds into Euclidean spaces with group actions.
method The paper provides upper and lower bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
result Explicit bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
We will provide a lower bound for the equivariant Lusternik-Schnirelmann category of an arbitrary proper action in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant Lusternik-Schnirelmann category of its generalized Weyl group. As an application we repro…
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.
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.
New bounds on knot unknotting numbers using involutive homology.
problem Bounding the unknotting number of strongly invertible knots.
method Using involutive Bar-Natan homology to establish bounds.
result Identified knots with strict inequality between standard and equivariant unknotting numbers.
Study on knots, genera, and algebraic concordance groups.
problem Understanding the equivariant slice genus of strongly invertible knots.
method Using the Blanchfield form to establish lower bounds and formulate an equivariant algebraic concordance group.
result The equivariant slice genus of an equivariant connected sum of a genus one strongly invertible slice knot is at least n/4.
New invariants refine link homology, showing large genus differences.
problem Understanding genus differences in equivariant cobordisms.
method Refined Bar-Natan homology for involutive links, constructing new numerical invariants.
result Difference between equivariant and isotopy-equivariant slice genera can be arbitrarily large.
The study examines how weight sharing, equivariance, and locality affect the sample complexity of neural networks.
problem Understanding the impact of design choices on the generalization error of neural networks.
method Statistical learning theory applied to single hidden layer networks with weight sharing, equivariance, and locality.
result Lower and upper bounds for sample complexity are derived, showing that locality has benefits but comes with a trade-off.
We show that the S^1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S^1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S^1-actio…
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d-invariants with Froyshov-type inequality. New invariants from Seiberg-Witten theory for 3-spheres with involution.
problem Equivariant Seiberg-Witten Floer theory of rational homology 3-spheres.
method Coupling involution to Seiberg-Witten theory, constructing delta-invariants.
result New Floer-theoretic invariants with properties and applications.
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.
Let M2n be a unitary torus (2n)-manifold, i.e., a (2n)-dimensional oriented stable complex connected closed Tn-manifold having a nonempty fixed set. In this paper we show that M bounds equivariantly if and only if the equivariant Chern numbers <(c1Tn)i(c2Tn)j,[M]>=0 for all $i, j\in {\Bbb …
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. 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.
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. New framework for conformal equivariant cycles in KK-theory.
problem Tackles conformal equivariance in unbounded KK-theory.
method Extends unbounded Kasparov theory with novel perturbation theory.
result Defines new unbounded representatives of Kasparov classes.
Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.
problem Embedding Hermitian symmetric spaces into their tangent spaces.
method Using polarity of the K-action to construct equivariant embeddings.
result Characterizes holomorphic/symplectic embeddings and realizes submanifolds.
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.
New minimal hypersphere found in 4-sphere solving Bernstein problem.
problem Spherical Bernstein problem in S4 method Equivariant min-max theory for G-invariant minimal hypersurfaces result Construction of embedded non-equatorial minimal hypersphere
The paper finds a special hypersurface in a manifold with positive Ricci curvature.
problem Finding a special hypersurface in a manifold with positive Ricci curvature.
method Equivariant min-max method applied to G-manifolds. result The hypersurface is a multiplicity one minimal G-hypersurface. The paper investigates how symmetry in models affects their performance and generalization.
problem Understanding how symmetry in models impacts their performance and generalization.
method Formal unified investigation of intuitions about symmetry in models and data.
result Quantitative bounds and comparisons between model and data equivariance lead to optimal model performance.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of G-invariant minimal hypersurfaces. method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
Our work improves VAE latent space clustering by enforcing invariant and equivariant learning.
problem Current VAEs fail to learn invariant and equivariant clusters in latent space.
method We use a mixture model pdf like Gaussian mixtures to enforce deep, group-invariant learning and separate semantic and equivariant variables.
result Our model effectively learns to disentangle invariant and equivariant representations, improving learning rate and image recognition.
We show a non-existence result for some class of equivariant maps between sphere bundles over tori. The notion of equivariant KO-degree is used in the proof. As an application to Seiberg-Witten theory, for a connected closed oriented spin 4-manifold with indefinite intersection form, we have a new bound of the second B…
Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov--Hausdorff topology, then the limit space is equivariantly …
For covering spaces and properly discontinuous actions with compatible diffusion operators, we discuss Lyons-Sullivan discretizations of the associated diffusions and harmonic functions of bounded growth.
Let G a semisimple Lie group of non-compact type and let XG be the Riemannian symmetric space associated to it. Suppose XG has dimension n and it has no factor isometric to either H2 or SL(3,R)/SO(3). Given a closed n-dimensional Riemannian manifold $…
New Wasserstein divergence improves generative model robustness and structure preservation.
problem Improving generative model robustness and structure preservation.
method Introduces a novel Wasserstein-1 path-space divergence and a WUP theorem.
result Derives robustness and generalization bounds for flow-based models.
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
New results on relative simplicial volume using bounded acyclicity.
problem Understanding relative simplicial volume in bounded cohomology.
method Equivariant nerve pairs, relative classifying spaces, and small relative amenable category.
result Vanishing results for ℓ2-Betti numbers and mapping degrees. Study proves obstructions to equivariantly slice strongly negative amphichiral knots.
problem Proving obstructions for equivariantly slice strongly negative amphichiral knots.
method Using determinant, Spinc-structures, Donaldson's theorem, and Heegaard Floer correction terms.
result 8 out of 16 strongly negative amphichiral knots with 12 or fewer crossings are not equivariantly slice.
New knot concordance invariants from Seiberg-Witten theory bound slice genus.
problem Bounding the slice genus of knots.
method Equivariant Seiberg-Witten-Floer cohomology applied to cyclic covers.
result Lower bounds on slice genus from knot concordance invariants.
The paper calculates the equivariant genus for a specific type of knot.
problem Calculating the equivariant genus of marked strongly invertible knots associated with 2-bridge knots.
method Analyzing invariant Seifert surfaces for marked strongly invertible knots.
result The paper completely determines the equivariant genus for every marked strongly invertible knot with K a 2-bridge knot.