Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.

翻译：对称性学习已被证明是提取数据隐藏结构的有效方法，其中等变关系概念发挥着核心作用。然而，当前大多数研究都建立在架构理论及对数据形式的相应假设之上。我们提出了神经傅里叶变换（Neural Fourier Transform，NFT）——一种无需预先假定群在数据上如何作用即可学习群潜在线性作用的通用框架。我们给出了NFT的理论基础，并证明了等变学习中普遍假设存在的线性等变特征，等价于数据空间上存在群不变核。此外，我们还提供了实验结果，以展示NFT在关于作用群知识层次不同的典型场景中的应用。