Datasets often have their intrinsic symmetries, and particular deep-learning
models called equivariant or invariant models have been developed to exploit
these symmetries. However, if some or all of these symmetries are only
approximate, which frequently happens in practice, these models may be
suboptimal due to the architectural restrictions imposed on them. We tackle
this issue of approximate symmetries in a setup where symmetries are mixed,
i.e., they are symmetries of not single but multiple different types and the
degree of approximation varies across these types. Instead of proposing a new
architectural restriction as in most of the previous approaches, we present a
regularizer-based method for building a model for a dataset with mixed
approximate symmetries. The key component of our method is what we call
equivariance regularizer for a given type of symmetries, which measures how
much a model is equivariant with respect to the symmetries of the type. Our
method is trained with these regularizers, one per each symmetry type, and the
strength of the regularizers is automatically tuned during training, leading to
the discovery of the approximation levels of some candidate symmetry types
without explicit supervision. Using synthetic function approximation and motion
forecasting tasks, we demonstrate that our method achieves better accuracy than
prior approaches while discovering the approximate symmetry levels correctly.

本文提出了一种基于正则化的方法，用于在具有混合近似对称性的数据集上构建模型，通过该方法可以依据每种对称性类型的等变正则化器训练模型，自动调整正则化器的强度，从而发现某些候选对称性类型的近似水平，同时在功能拟合和运动预测任务中展示了比以前的方法更高的精度。

混合对称下的软等变正则化

Regularizing Towards Soft Equivariance Under Mixed Symmetries

Tensor factorization and distanced based models play important roles in
knowledge graph completion (KGC). However, the relational matrices in KGC
methods often induce a high model complexity, bearing a high risk of
overfitting. As a remedy, researchers propose a variety of different
regularizers such as the tensor nuclear norm regularizer. Our motivation is
based on the observation that the previous work only focuses on the "size" of
the parametric space, while leaving the implicit semantic information widely
untouched. To address this issue, we propose a new regularizer, namely,
Equivariance Regularizer (ER), which can suppress overfitting by leveraging the
implicit semantic information. Specifically, ER can enhance the generalization
ability of the model by employing the semantic equivariance between the head
and tail entities. Moreover, it is a generic solution for both distance based
models and tensor factorization based models. The experimental results indicate
a clear and substantial improvement over the state-of-the-art relation
prediction methods.

通过引入 Equivariance Regularizer 作为一种新的约束方法，可以通过利用 head 和 tail 实体之间的语义等变性来提高模型的泛化能力，从而有效解决了当前方法中存在的大小参数空间问题，并取得了实验结果上的显着提高。