The homology groups of many natural sequences of groups (e.g. general linear groups, mapping class groups, etc.) stabilize as . Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequ…
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48 results for “Representation stability”
New stability result for Milnor fibers of pure braid groups.
problem Understanding the stable homology of Milnor fibers.
method Representation stability in the context of Milnor fibers of pure braid groups.
result Computed stable integral homology of Milnor fibers.
We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical t…
Study on disk configurations in strips shows stability patterns.
problem Understanding stability patterns in disk configurations in strips.
method Finite presentation of rational homology groups, representation stability.
result Disk configuration space exhibits first-order representation stability.
Geometric stability measures neural network robustness, distinguishing from similarity metrics.
problem Lack of robustness in neural network representations.
method Introduces geometric stability, quantified by Shesha metric measuring self-consistency.
result Stability and similarity are uncorrelated, revealing distinct properties of neural network robustness.
Stability in cohomology of linear subspace arrangements.
problem Stability in cohomology of linear subspace arrangements.
method FI-modules, representation stability, generalized character polynomials.
result Cohomology of linear subspace arrangements satisfies strong form of representation stability.
The paper proves stabilization in hypersurface sections using Grothendieck rings and probabilistic methods.
problem Stabilization of configuration spaces and Hodge Euler characteristics for hypersurface sections.
method Geometric and cohomological stabilization, probabilistic interpretation, Grothendieck ring analysis.
result Explicit formulas for stable values of Hodge Euler characteristics and point counts.
Stabilizes cohomology of configuration spaces over complex varieties.
problem Stability of cohomology of configuration spaces.
method Probabilistic interpretation of stabilization as asymptotic independence of motivic random variables.
result Explicit formulas for limits in terms of motivic Euler product.
Representation Stabilitymath.GT
Representation stability is a phenomenon whereby the structure of certain sequences of spaces can be seen to stabilize when viewed through the lens of representation theory. In this paper I describe this phenomenon and sketch a framework, the theory of FI-modules, that explains the mechanism behind it.
The paper shows stable behavior in cohomology of configuration spaces.
problem Stability of cohomology of configuration spaces in R^d.
method Analysis of S_n-representations controlling cohomology.
result Representation stability in cohomology of configuration spaces stabilizes sharply.
New method proves representation stability for linear groups, resolving homological questions.
problem Proving representation stability for linear groups over fields of characteristic zero.
method Introducing a technique for quantitative representation stability theorems.
result Vanishing result for higher syzygies of VIC- and SI-modules.
The paper proves stability of martingale representations in a broad context.
problem Stability of martingale representations in a general framework.
method Extensive use of martingale theory and convergence properties.
result Each component of the martingale representation converges under Skorokhod topology.
Enhances stability ranges for Torelli and congruence subgroup homologies.
problem Improving stability ranges for specific subgroup homologies.
method Analyzes H2(Torelli subgroup of Aut(Fn)'s), H2(Torelli subgroup of mapping class groups), and Hk(congruence subgroups of GL_n(R)'s).
result Improved central stability ranges for various subgroup homologies.
Neural nets learn robust geometric data representations.
problem Ensuring neural networks are robust to adversarial attacks.
method Topological Data Analysis via persistence diagrams, Lipschitz stability.
result Certified -robustness on ORBIT5K dataset.
The paper explores how the cohomology of certain space arrangements stabilizes as the number of subspaces increases.
problem Stability of cohomology groups of complements of linear subspace arrangements.
method Representation stability in the context of cohomology groups, focusing on arrangements invariant under permutation of coordinates.
result Bounds on stabilization and alternative proof for the stabilization of cohomology groups.
Central stability for the homology of congruence subgroups and the second homology of Torelli groupsmath.AT
New stability results for homology groups of Torelli subgroups and congruence subgroups.
problem Stability of homology groups of Torelli subgroups and congruence subgroups.
method Representation stability and syzygies of modules with finite polynomial degree.
result New stability results for the second homology groups of Torelli subgroups and homology of certain congruence subgroups.
Study stabilizes arithmetic statistics of rational maps over finite fields.
problem Stability of arithmetic statistics of rational maps over finite fields.
method Representation stability and arithmetic statistics of spaces of 0-cycles.
result Arithmetic quantities associated to rational maps over finite fields stabilize as degree increases.
Unified approach to representation stability and character polynomials.
problem Generalizing representation stability across different groups.
method Axiomatic approach to categories of FI type.
result New types of categories (e.g. ) that exhibit stabilization.
The paper proves representation stability for Torelli groups and their filtrations.
problem Representation stability of Torelli groups and their filtrations.
method Uniform representation stability using rational -modules and -modules.
result The quotients of the lower central series of Torelli subgroups are uniformly representation stable.
Study of modular representations in homology of congruence subgroups.
problem Understanding modular representations in homology of congruence subgroups.
method Analysis of sequences of modular representations of symplectic and special linear groups over finite fields.
result Established periodic representation stability in the sense of Church--Farb.
Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the un…
Study max- and min-stability under first-order stochastic dominance, finding new functional characterizations.
problem Understanding max- and min-stability in stochastic dominance.
method Representation theorem for functionals satisfying max-stability, combining max- and min-stability to define Lambda-quantiles.
result New characterizations of functionals, including Lambda-quantiles, in finance and political science.
The paper shows how to stabilize off-policy reinforcement learning using specific state representations.
problem Stability issues in reinforcement learning with function approximation and off-policy learning.
method Formal analysis of representation learning schemes based on the transition matrix of a policy.
result Schur and orthogonal bases of the Krylov subspace provide stable representations for TD learning.
SIGNNAP learns stable and identifiable node representations in GNNs against graph perturbations.
problem Fragility of GNN models to graph perturbations leading to unreliable node representations.
method SIGNNAP proposes a novel model that learns stable and identifiable node representations in an unsupervised manner, formalizing stability and identifiability through a contrastive objective and preserving smoothness with existing GNN backbones.
result SIGNNAP demonstrates effectiveness in learning stable and identifiable node representations in GNNs against graph perturbations on six benchmarks.
Stability in homology of moduli spaces of admissible covers.
problem Stability of homology groups of moduli spaces of admissible covers.
method Representation stability using combinatorial category structure.
result Rational generating function for homology ranks with specified poles.
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
The paper connects arithmetic statistics to homological stability in étale cohomology.
problem Understanding the growth of arithmetic statistics over finite fields.
method Importing homological stability from topology to étale cohomology.
result Established subexponential bounds on the growth of unstable cohomology.
Sparse representations improve network robustness and stability.
problem The benefits of sparse representations in artificial networks.
method Analysis of sparse networks with sparse weights and activations, simulations on MNIST and Google Speech Command Dataset.
result Sparse networks show significantly improved robustness and stability compared to dense networks.
The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.
problem Understanding the homology of ordered configuration spaces of graphs.
method Explicit formulas for asymptotic Betti numbers and homology multiplicities in characteristic zero.
result Explicit formulas for asymptotic multiplicities in homology of irreducible representations of the symmetric group.
We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representatio…
KCRL learns stable policies for nonlinear systems with formal guarantees.
problem Lack of stabilization guarantees in RL methods for safety-critical systems.
method KCRL uses Krasovskii's Lyapunov functions as a stability constraint and a primal-dual approach to learn stabilizing policies.
result KCRL guarantees learning a stabilizing policy in a finite number of interactions.
Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representationsstat.ML
The paper analyzes deep convolutional representations and their properties.
problem Understanding the properties of deep convolutional architectures.
method Introducing a multilayer kernel based on convolutional kernel networks and studying the geometry induced by the kernel mapping.
result Characterization of the RKHS and its relation to model complexity and generalization.
Scattering transforms adapted for non-Euclidean domains using diffusion wavelets.
problem Stability of data representations in non-Euclidean domains.
method Generalization of scattering transforms to non-Euclidean domains using diffusion wavelets and diffusion maps.
result Stability of the representation to metric perturbations of the domain.
Study on cohomology of level 4 braid group, proving representation stability.
problem Investigating cohomology of level 4 braid group.
method Exact formula for first Betti number, representation stability, representation theory development.
result Representation stability in level 4 braid group.
New models learn stable latent clusters without side info.
problem Stability of non-linear ICA representations without side information.
method Deep generative models with latent clusterings, compared to standard VAEs and auxiliary labeled models.
result Deep generative models with latent clusterings are as stable as models with side information.
Spectral graph sparsification preserves geometry of GNN embeddings.
problem Maintaining geometric properties of graph neural network embeddings during sparsification.
method Proving spectral sparsification preserves squared pairwise distances, class means, and covariance structure in embedding space.
result Spectral sparsification preserves the geometry of learned embeddings in GNNs.
Let M_g^n be the moduli space of Riemann surfaces of genus g with n labeled marked points. We prove that, for g \geq 2, the cohomology groups {H^i(M_g^n;Q)}_{n=1}^{\infty} form a sequence of Sn representations which is representation stable in the sense of Church-Farb [CF]. In particular this result applied to the triv…
AUASE embeds dynamic networks with stability guarantees for node comparison.
problem Stability in dynamic network embeddings for comparing nodes across time.
method Attributed unfolded adjacency spectral embedding (AUASE) for stable unsupervised learning.
result AUASE provides significant improvements in link prediction and node classification.
The paper studies mapping class group actions on character varieties of surfaces.
problem Understanding the dynamics of mapping class group actions on relative -character varieties.
method Definition and proof of simple-stability and primitive-stability of representations.
result Holonomies of hyperbolic cone surfaces are simple-stable and primitive-stable.
MuZero visualizes its internal representations to stabilize planning.
problem Stability issues in MuZero's planning process.
method Visualized MuZero's latent representations and proposed regularization techniques.
result Action trajectories diverge between observation embeddings and internal state transitions, leading to instability.
The abstract discusses homological stability in topological moduli spaces.
problem Homological stability in topological moduli spaces.
method Constructing canonical resolutions and introducing coefficient systems.
result Homological stability for various moduli spaces.
The EM algorithm's convergence is analyzed using Lyapunov stability theory.
problem Analyzing the convergence of the EM algorithm.
method Reinterpreting the EM algorithm as a dynamical system and applying Lyapunov stability theory.
result Asymptotic stability and convergence of the EM algorithm are established.
Short proof shows Borel class stability under Cartan involution for 3-manifold groups.
problem Stability of Borel class under Cartan involution for 3-manifold representations.
method Direct proof for PGL(3,C) representations, leveraging a more general result for PGL(n,C).
result Borel class is preserved under Cartan involution up to sign for 3-manifold groups.
Simplified proof of equivalence between stability and Bowditch conditions.
problem Equivalence of stability and Bowditch conditions in free group representations.
method Simplified proof using Bowditch's tree language and ideas from Lee and Xu.
result Equivalence of primitive stability and Bowditch conditions proved.
Study on representations of four-punctured sphere group in hyperbolic spaces.
problem Understanding representations of the four-punctured sphere group.
method Investigation into simple-stable and Bowditch representations in Gromov-hyperbolic spaces.
result Simple-stable representations and Bowditch representations are equivalent.
New stabilization method in graph braid homology yields polynomial growth.
problem Stabilization in graph braid homology.
method Introduced a stabilization map on graph configuration spaces, leading to a polynomial ring action on homology.
result Homology module is finitely generated and shows polynomial growth in Betti numbers.
New findings on stability and Q-conditions for free group actions in hyperbolic spaces.
problem Understanding stability and Q-conditions for free group actions in hyperbolic spaces.
method Generalization of Minsky's and Bowditch's results to higher dimensions and W_3-extensible representations.
result Equivalence between primitive stability and generalized Q-conditions for F_2 in hyperbolic d-space (d >= 3).