The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive operators (GENEOs), which are maps between function spaces…
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48 results for “Group Equivariant Non-Expansive Operators (GENEOs)”
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.
Proposes a topological model for partial equivariance in neural networks.
problem Capturing partial equivariance in neural networks for data analysis.
method Introduces P-GENEOs and studies spaces of measurements and P-GENEOs between them.
result Spaces of measurements and P-GENEOs have convenient approximation and convexity properties.
On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operatorsmath.DG
The paper studies geometric properties of group equivariant operators and their Riemannian structure.
problem Understanding the geometric structure of group equivariant operators.
method Endowing the space of group equivariant non-expansive operators with a Riemannian manifold structure and using gradient descent methods.
result Gradient descent methods can be applied to minimize cost functions on the space of group equivariant non-expansive operators.
SCENE-Net improves 3D point cloud segmentation with low resource usage and transparency.
problem Lack of resources and transparency in 3D semantic segmentation models.
method SCENE-Net uses signature shapes identified via GENEOs to achieve semantic segmentation with minimal resources.
result SCENE-Net achieves comparable IoU to state-of-the-art methods with less data and computational resources.
Constructs equivariant spectral flow for Dirac-type operators on manifolds.
problem Calculating spectral flow for Dirac-type operators on manifolds with group actions.
method Equivariant spectral flow construction for paths of Dirac-type operators on manifolds.
result Relates delocalised η-invariants and ρ-invariants for different positive scalar curvature metrics.
Formula for index in Lorentzian spacetimes.
problem Developing an index formula for spacetimes with boundary.
method Reduction from equivariant to non-equivariant, Lorentzian spectral flow.
result Equivalence of equivariant index and spectral flow in Lorentzian spacetimes.
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.
We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-va…
In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it …
New analysis of stochastic approximation with non-expansive mappings.
problem Finite-time analysis of two-time-scale stochastic approximation with non-expansive mappings.
method Studied two-time-scale stochastic approximation algorithms with non-expansive mappings and projection steps.
result Last-iterate mean square residual error decays at a rate .
Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvaturemath.DG
We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G…
Defines an equivariant index for proper actions by .
problem Generalizing index theory for proper actions by groups.
method Introduces an -equivariant index for -transversally elliptic operators.
result Generalizes Baum--Connes, Atiyah's index, and Kawasaki's orbifold index.
In this expository article, we consider first order elliptic differential operators acting on smooth vector bundles over compact manifolds, and certain invariants derived from the analysis of these operators, namely the eta invariant} and the equivariant index. Many researchers have previously considered these invarian…
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.
The study quantifies how many objects can be linearly classified under all views.
problem Understanding the expressivity of group-equivariant representations.
method Generalization of Cover's Function Counting Theorem to quantify separable dichotomies.
result The fraction of separable dichotomies is determined by the fixed space dimension of the group action.
Paper generalizes spectral flow formulas for compact Lie group actions.
problem Generalizing spectral flow formulas for compact Lie group actions.
method Equivariant version of Dai-Zhang higher spectral flow, embedding formula, adiabatic limit formula for Atiyah-Patodi-Singer eta invariants.
result Generalization of eta forms to equivariant Bismut-Cheeger eta forms.
Guillemin trace formula adapted for group actions.
problem Distributional trace for proper, cocompact group actions.
method Developing an equivariant version of the distributional trace.
result Equivariant Guillemin trace formula for group actions.
We define an equivariant index of Spin-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to the quantisation commutes with reduction principle.
Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actionscs.LG
EbC learns equivariant embeddings from unlabeled group actions.
problem Learning equivariant embeddings from unlabeled group actions.
method Equivariance by Contrast (EbC) method to learn equivariant embeddings from observation pairs .
result High-fidelity equivariance in latent space for diverse groups.
Equivariant indices have previously been defined in cases where either the group or the orbit space in question is compact. In this paper, we develop an equivariant index without assuming the group or the orbit space to be compact. This allows us to generalise an index of deformed Dirac operators, defined for compact g…
Maximal concentration bounds for stochastic approximation with heavy-tailed noise.
problem Analyzing the convergence of stochastic approximation algorithms under heavy-tailed Markovian noise.
method Novel Lyapunov function and black-box truncation argument.
result Tail behavior of the error can be sub-Gaussian, sub-Weibull, or lighter than any Pareto but heavier than any Weibull.
We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a -manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formu…
We derive a formula for the eta invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extension. As an example, we give some computations for spheres.
Classifies -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.
Paper calculates indices for group actions using cocycles.
problem Calculating equivariant indices for group actions.
method Construct cyclic cocycles on Harish-Chandra's Schwartz algebra, compute pairings with equivariant indices.
result Index formula completely determines equivariant indices via topological expressions.
A strong from of invariance under a group G is manifested in a family over the classifying space BG. We advocate a differential-geometric avatar of BG when G is a Lie group. Applied to G-equivariant connections on smooth principal or vector bundles, the equivariance-->families principle converts the G-equivariant exten…
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…
Geometric models improve feature extraction and equivariance in image generation.
problem Improving feature extraction at multiscale levels and reducing network complexity.
method Proposes a geometric generative model based on morphological PDEs and GANs, incorporating equivariance for geometric interpretability.
result Preliminary results show GM-GAN outperforms classical GANs on MNIST data.
Formula derived for Dirac operators on Lie groupoids.
problem Equivariant Dirac operators on Lie groupoids.
method Getzler rescaling method to derive a fixed-point formula.
result Reduces to standard fixed-point formula for closed manifolds.
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.
Frame Averaging makes neural networks invariant or equivariant to new symmetries.
problem Designing neural networks that respect symmetries while being expressive and efficient.
method Introduces Frame Averaging (FA) as a systematic framework to adapt architectures to become invariant or equivariant to new symmetries.
result Frame Averaging guarantees exact invariance or equivariance while being simpler to compute than full group averaging.
In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivaria…
A softmax operator applied to a set of values acts somewhat like the maximization function and somewhat like an average. In sequential decision making, softmax is often used in settings where it is necessary to maximize utility but also to hedge against problems that arise from putting all of one's weight behind a sing…
Let be a compact connected Lie group, and a compact Hamiltonian -space, with moment map . For each -equivariant Hermitian vector bundle over , one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of . In the present paper, we study gluing prop…
We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe a…
Value function estimation is an important task in reinforcement learning, i.e., prediction. The Boltzmann softmax operator is a natural value estimator and can provide several benefits. However, it does not satisfy the non-expansion property, and its direct use may fail to converge even in value iteration. In this pape…
Study index theory on Lie group homogeneous spaces using topological and analytic methods.
problem Index theory on homogeneous spaces of Lie groups.
method Topological and analytic approaches: Riemann-Roch formula and heat kernel methods.
result Local index formula representing higher indices of equivariant elliptic operators.
A global view of equivariant vector bundles and Dirac operators on some compact homogeneous spacesmath.DG
In order to facilitate the comparison of Riemannian homogeneous spaces of compact Lie groups with noncommutative geometries ("quantizations") that approximate them, we develop here the basic facts concerning equivariant vector bundles and Dirac operators over them in a way that uses only global constructions and argume…
In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equiva…
In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a -manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applicatio…
Absolute index theorem for warped product manifolds.
problem Equivariant index computation for manifolds with warped product structures.
method Warped product structure, Fredholm operator, Atiyah-Segal-Singer index theorem.
result Equivariant relative index theorem for manifolds with warped product structures.
New framework uses symmetry-based matrices for efficient, flexible NNs.
problem Designing neural networks with relaxed equivariance.
method Symmetry-based structured matrices, Group Matrices (GMs).
result GMs enable competitive performance with fewer parameters.
We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original p…
The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.
The Geometry of Filteringmath.DG
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …
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.
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.