A Characterization Theorem for Equivariant Networks with Point-wise Activations

• URL: http://arxiv.org/abs/2401.09235v1
• Date: Wed, 17 Jan 2024 14:30:46 GMT
• Title: A Characterization Theorem for Equivariant Networks with Point-wise Activations
• Authors: Marco Pacini, Xiaowen Dong, Bruno Lepri and Gabriele Santin
• Abstract summary: We prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. We show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.
• Score: 13.00676132572457
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Equivariant neural networks have shown improved performance, expressiveness and sample complexity on symmetrical domains. But for some specific symmetries, representations, and choice of coordinates, the most common point-wise activations, such as ReLU, are not equivariant, hence they cannot be employed in the design of equivariant neural networks. The theorem we present in this paper describes all possible combinations of finite-dimensional representations, choice of coordinates and point-wise activations to obtain an exactly equivariant layer, generalizing and strengthening existing characterizations. Notable cases of practical relevance are discussed as corollaries. Indeed, we prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. Then, we discuss implications of our findings when applied to important instances of exactly equivariant networks. First, we completely characterize permutation equivariant networks such as Invariant Graph Networks with point-wise nonlinearities and their geometric counterparts, highlighting a plethora of models whose expressive power and performance are still unknown. Second, we show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.

Related papers

• Equivariant neural networks and piecewise linear representation theory [0.0]
  Equivariant neural networks are neural networks with symmetry. Motivated by the theory of group representations, we decompose the layers of an equivariant neural network into simple representations.
  arXiv  Detail & Related papers  (2024-08-01T23:08:37Z)
• Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations [111.27636893711055]
  We study the filtering and stability properties of non commutative convolutional filters. Our results have direct implications for group neural networks, multigraph neural networks and quaternion neural networks.
  arXiv  Detail & Related papers  (2023-10-05T20:27:22Z)
• Geometry of Linear Neural Networks: Equivariance and Invariance under Permutation Groups [0.0]
  We investigate the subvariety of functions that are equivariant or invariant under the action of a permutation group. We draw conclusions for the parameterization and the design of equivariant and invariant linear networks.
  arXiv  Detail & Related papers  (2023-09-24T19:40:15Z)
• Using and Abusing Equivariance [10.70891251559827]
  We show how Group Equivariant Convolutional Neural Networks use subsampling to learn to break equivariance to their symmetries. We show that a change in the input dimension of a network as small as a single pixel can be enough for commonly used architectures to become approximately equivariant, rather than exactly.
  arXiv  Detail & Related papers  (2023-08-22T09:49:26Z)
• Optimization Dynamics of Equivariant and Augmented Neural Networks [2.7918308693131135]
  We investigate the optimization of neural networks on symmetric data. We compare the strategy of constraining the architecture to be equivariant to that of using data augmentation. Our analysis reveals that even in the latter situation, stationary points may be unstable for augmented training although they are stable for the manifestly equivariant models.
  arXiv  Detail & Related papers  (2023-03-23T17:26:12Z)
• Generalization capabilities of neural networks in lattice applications [0.0]
  We investigate the advantages of adopting translationally equivariant neural networks in favor of non-equivariant ones. We show that our best equivariant architectures can perform and generalize significantly better than their non-equivariant counterparts.
  arXiv  Detail & Related papers  (2021-12-23T11:48:06Z)
• Revisiting Transformation Invariant Geometric Deep Learning: Are Initial Representations All You Need? [80.86819657126041]
  We show that transformation-invariant and distance-preserving initial representations are sufficient to achieve transformation invariance. Specifically, we realize transformation-invariant and distance-preserving initial point representations by modifying multi-dimensional scaling. We prove that TinvNN can strictly guarantee transformation invariance, being general and flexible enough to be combined with the existing neural networks.
  arXiv  Detail & Related papers  (2021-12-23T03:52:33Z)
• Geometric and Physical Quantities improve E(3) Equivariant Message Passing [59.98327062664975]
  We introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks. This model, composed of steerables, is able to incorporate geometric and physical information in both the message and update functions. We demonstrate the effectiveness of our method on several tasks in computational physics and chemistry.
  arXiv  Detail & Related papers  (2021-10-06T16:34:26Z)
• Topographic VAEs learn Equivariant Capsules [84.33745072274942]
  We introduce the Topographic VAE: a novel method for efficiently training deep generative models with topographically organized latent variables. We show that such a model indeed learns to organize its activations according to salient characteristics such as digit class, width, and style on MNIST. We demonstrate approximate equivariance to complex transformations, expanding upon the capabilities of existing group equivariant neural networks.
  arXiv  Detail & Related papers  (2021-09-03T09:25:57Z)
• Convolutional Filtering and Neural Networks with Non Commutative Algebras [153.20329791008095]
  We study the generalization of non commutative convolutional neural networks. We show that non commutative convolutional architectures can be stable to deformations on the space of operators.
  arXiv  Detail & Related papers  (2021-08-23T04:22:58Z)
• Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs [81.12344211998635]
  A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs) We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods.
  arXiv  Detail & Related papers  (2020-03-11T17:21:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.