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：近期，模型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。对现有神经网络架构内在等变性的分析，以及显式"内置"等变性的模型构建研究，已发展为独立的重要研究领域。然而，为架构赋予特定群等变性，会使模型对预期遇到的数据变换类型形成强先验假设。严格等变模型虽然强制满足对称性，但现实数据往往并不遵循如此严格的等变约束。在这种情况下，严格等变的先验可能过于强烈，反而导致模型性能不佳。因此，本文研究与之密切相关的"近似等变性"课题。我们给出了近似等变性的定义，并通过引入李群的李代数，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，并证明其相比李群卷积具有若干优势，包括适用于具有非满射指数映射的非紧李群。在此基础上，我们揭示了等变性与等距性、近似等变性与近似等距性之间的关联。我们证明了两条存在性定理：其一证明了在流形等距映射的有界距离内存在近似等距映射，其二证明了希尔伯特空间中该命题的逆命题也成立。我们将这些定理推广到存在性证明：在群作用与函数类满足特定约束条件下，完全等变嵌入函数的有界距离内存在近似等变流形嵌入。最后，通过在完全等变与近似等变设定下对数据集进行基准测试，验证了本方法的有效性。