Almost Equivariance via Lie Algebra Convolutions

• URL: http://arxiv.org/abs/2310.13164v6
• Date: Thu, 20 Jun 2024 03:08:14 GMT
• Title: Almost Equivariance via Lie Algebra Convolutions
• Authors: Daniel McNeela,
• Abstract summary: We provide a definition of almost equivariance and give a practical method for encoding it in models. Specifically, we define Lie algebra convolutions and demonstrate that they offer several benefits over Lie group convolutions. We prove two existence theorems, one showing the existence of almost isometries within bounded distance of isometries of a manifold, and another showing the converse for Hilbert spaces.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recently, the equivariance of models with respect to a group action has become an important topic of research in machine learning. Analysis of the built-in equivariance of existing neural network architectures, as well as the study of building models that explicitly "bake in" equivariance, have become significant research areas in their own right. However, imbuing an architecture with a specific group equivariance imposes a strong prior on the types of data transformations that the model expects to see. While strictly-equivariant models enforce symmetries, real-world data does not always conform to such strict equivariances. In such cases, the prior of strict equivariance can actually prove too strong and cause models to underperform. Therefore, in this work we study a closely related topic, that of almost equivariance. We provide a definition of almost equivariance and give a practical method for encoding almost equivariance in models by appealing to the Lie algebra of a Lie group. Specifically, we define Lie algebra convolutions and demonstrate that they offer several benefits over Lie group convolutions, including being well-defined for non-compact Lie groups having non-surjective exponential map. From there, we demonstrate connections between the notions of equivariance and isometry and those of almost equivariance and almost isometry. We prove two existence theorems, one showing the existence of almost isometries within bounded distance of isometries of a manifold, and another showing the converse for Hilbert spaces. We extend these theorems to prove the existence of almost equivariant manifold embeddings within bounded distance of fully equivariant embedding functions, subject to certain constraints on the group action and the function class. Finally, we demonstrate the validity of our approach by benchmarking against datasets in fully equivariant and almost equivariant settings.

Related papers

• Lie Group Decompositions for Equivariant Neural Networks [12.139222986297261]
  We show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the benchmark affine-invariant classification task.
  arXiv  Detail & Related papers  (2023-10-17T16:04:33Z)
• Approximation-Generalization Trade-offs under (Approximate) Group Equivariance [3.0458514384586395]
  Group equivariant neural networks have demonstrated impressive performance across various domains and applications such as protein and drug design. We show how models capturing task-specific symmetries lead to improved generalization. We examine the more general question of model mis-specification when the model symmetries don't align with the data symmetries.
  arXiv  Detail & Related papers  (2023-05-27T22:53:37Z)
• A General Theory of Correct, Incorrect, and Extrinsic Equivariance [22.625954325866907]
  A missing piece in the equivariant learning literature is the analysis of equivariant networks when symmetry exists only partially in the domain. In this work, we present a general theory for such a situation. We prove error lower bounds for invariant or equivariant networks in classification or regression settings with partially incorrect symmetry.
  arXiv  Detail & Related papers  (2023-03-08T17:28:54Z)
• The Lie Derivative for Measuring Learned Equivariance [84.29366874540217]
  We study the equivariance properties of hundreds of pretrained models, spanning CNNs, transformers, and Mixer architectures. We find that many violations of equivariance can be linked to spatial aliasing in ubiquitous network layers, such as pointwise non-linearities. For example, transformers can be more equivariant than convolutional neural networks after training.
  arXiv  Detail & Related papers  (2022-10-06T15:20:55Z)
• Equivariant Disentangled Transformation for Domain Generalization under Combination Shift [91.38796390449504]
  Combinations of domains and labels are not observed during training but appear in the test environment. We provide a unique formulation of the combination shift problem based on the concepts of homomorphism, equivariance, and a refined definition of disentanglement.
  arXiv  Detail & Related papers  (2022-08-03T12:31:31Z)
• On the Strong Correlation Between Model Invariance and Generalization [54.812786542023325]
  Generalization captures a model's ability to classify unseen data. Invariance measures consistency of model predictions on transformations of the data. From a dataset-centric view, we find a certain model's accuracy and invariance linearly correlated on different test sets.
  arXiv  Detail & Related papers  (2022-07-14T17:08:25Z)
• Equivariance Discovery by Learned Parameter-Sharing [153.41877129746223]
  We study how to discover interpretable equivariances from data. Specifically, we formulate this discovery process as an optimization problem over a model's parameter-sharing schemes. Also, we theoretically analyze the method for Gaussian data and provide a bound on the mean squared gap between the studied discovery scheme and the oracle scheme.
  arXiv  Detail & Related papers  (2022-04-07T17:59:19Z)
• Commutative Lie Group VAE for Disentanglement Learning [96.32813624341833]
  We view disentanglement learning as discovering an underlying structure that equivariantly reflects the factorized variations shown in data. A simple model named Commutative Lie Group VAE is introduced to realize the group-based disentanglement learning. Experiments show that our model can effectively learn disentangled representations without supervision, and can achieve state-of-the-art performance without extra constraints.
  arXiv  Detail & Related papers  (2021-06-07T07:03:14Z)
• Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data [52.78581260260455]
  We propose a general method to construct a convolutional layer that is equivariant to transformations from any specified Lie group. We apply the same model architecture to images, ball-and-stick molecular data, and Hamiltonian dynamical systems.
  arXiv  Detail & Related papers  (2020-02-25T17:40:38Z)

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.