Constructing Invariant and Equivariant Operations by Symmetric Tensor Network

• URL: http://arxiv.org/abs/2508.12596v1
• Date: Mon, 18 Aug 2025 03:13:08 GMT
• Title: Constructing Invariant and Equivariant Operations by Symmetric Tensor Network
• Authors: Meng Zhang, Chao Wang, Hao Zhang, Shaojun Dong, Lixin He,
• Abstract summary: This work presents a systematic method for constructing valid invariant and equivariant operations.<n>It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types.<n>We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the law of materials.
• Score: 8.759616567360537
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

Related papers

• Permutation Equivariant Neural Networks for Symmetric Tensors [0.0]
  We present two different characterisations of all linear permutation equivariant functions between symmetric power spaces of $mathbbRn$.<n>We show that these functions are highly efficient compared to standard tensors and have potential to generalise well to symmetrics of different sizes.
  arXiv  Detail & Related papers  (2025-03-14T10:33:13Z)
• A Characterization Theorem for Equivariant Networks with Point-wise Activations [13.00676132572457]
  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.
  arXiv  Detail & Related papers  (2024-01-17T14:30:46Z)
• Permutation Equivariant Neural Functionals [92.0667671999604]
  This work studies the design of neural networks that can process the weights or gradients of other neural networks. We focus on the permutation symmetries that arise in the weights of deep feedforward networks because hidden layer neurons have no inherent order. In our experiments, we find that permutation equivariant neural functionals are effective on a diverse set of tasks.
  arXiv  Detail & Related papers  (2023-02-27T18:52:38Z)
• Oracle-Preserving Latent Flows [58.720142291102135]
  We develop a methodology for the simultaneous discovery of multiple nontrivial continuous symmetries across an entire labelled dataset. The symmetry transformations and the corresponding generators are modeled with fully connected neural networks trained with a specially constructed loss function. The two new elements in this work are the use of a reduced-dimensionality latent space and the generalization to transformations invariant with respect to high-dimensional oracles.
  arXiv  Detail & Related papers  (2023-02-02T00:13:32Z)
• Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism [12.008737454250463]
  We propose using fusion diagrams, a technique widely employed in simulating SU($2$)-symmetric quantum many-body problems, to design new equivariant components for equivariant neural networks. When applied to particles within a given local neighborhood, the resulting components, which we term "fusion blocks," serve as universal approximators of any continuous equivariant function. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.
  arXiv  Detail & Related papers  (2022-11-14T16:06:59Z)
• Learning Symmetric Embeddings for Equivariant World Models [9.781637768189158]
  We propose learning symmetric embedding networks (SENs) that encode an input space (e.g. images) This network can be trained end-to-end with an equivariant task network to learn an explicitly symmetric representation. Our experiments demonstrate that SENs facilitate the application of equivariant networks to data with complex symmetry representations.
  arXiv  Detail & Related papers  (2022-04-24T22:31:52Z)
• Equivariant vector field network for many-body system modeling [65.22203086172019]
  Equivariant Vector Field Network (EVFN) is built on a novel equivariant basis and the associated scalarization and vectorization layers. We evaluate our method on predicting trajectories of simulated Newton mechanics systems with both full and partially observed data.
  arXiv  Detail & Related papers  (2021-10-26T14:26:25Z)
• 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)
• Learning Equivariant Energy Based Models with Equivariant Stein Variational Gradient Descent [80.73580820014242]
  We focus on the problem of efficient sampling and learning of probability densities by incorporating symmetries in probabilistic models. We first introduce Equivariant Stein Variational Gradient Descent algorithm -- an equivariant sampling method based on Stein's identity for sampling from densities with symmetries. We propose new ways of improving and scaling up training of energy based models.
  arXiv  Detail & Related papers  (2021-06-15T01:35:17Z)
• Equivariant neural networks for inverse problems [1.7942265700058986]
  We show that group equivariant convolutional operations can naturally be incorporated into learned reconstruction methods. We design learned iterative methods in which the proximal operators are modelled as group equivariant convolutional neural networks.
  arXiv  Detail & Related papers  (2021-02-23T05:38:41Z)

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.