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.

翻译：近年来，模型相对于群作用的等变性已成为机器学习领域的重要研究方向。分析现有神经网络架构的内置等变性，以及构建显式"嵌入"等变性的模型，已发展成为独立的重要研究领域。然而，为架构赋予特定群等变性，会对模型预期处理的数据变换类型施加强先验。虽然严格等变模型强制要求对称性，但现实世界数据并不总是符合这种严格等变性。此时，严格等变先验可能过强，导致模型性能下降。因此，本文研究近似等变性这一密切相关的主题。我们给出了近似等变性的定义，并通过利用李群的李代数，提出了一种在模型中编码近似等变性的实用方法。具体而言，我们定义了李代数卷积，并论证其相比李群卷积具有多项优势，包括对具有非满射指数映射的非紧致李群也能良好定义。进而，我们揭示了等变性与等距性概念之间，以及近似等变性与近似等距性概念之间的联系。我们证明了两个存在性定理：其一表明在流形等距性的有界距离内存在近似等距性，其二给出了希尔伯特空间中的逆定理。我们将这些定理推广，证明了在群作用与函数类满足特定约束条件下，在全等变嵌入函数有界距离内存在近似等变流形嵌入。最后，通过在完全等变和近似等变设置下对数据集进行基准测试，验证了方法的有效性。