用于等变神经网络的李群分解
Lie Group Decompositions for Equivariant Neural Networks
AI总结：
针对现有李群等变神经网络方法受指数映射满射性限制、难以处理非紧非阿贝尔李群的问题，提出基于李群分解的框架，实现仿射等变模型构建，在基准任务上性能更优。
摘要：
几何变换的不变性与等变性已被证实是训练（卷积）神经网络模型时非常有效的归纳偏置，尤其是在小数据场景下。现有大量研究聚焦于所采用的对称群为紧群、阿贝尔群或兼具这两种性质的情况。近期研究探索将所用变换类别拓展至李群范畴，主要通过利用其李代数以及群指数映射和对数映射实现。这类方法的适用性受限于以下事实：根据目标群$G$的不同，指数映射可能不具备满射性。当$G$既非紧群也非阿贝尔群时，还会遇到更多限制。借助李群及其齐性空间的结构与几何性质，我们提出了一个可处理此类群的框架，主要聚焦于群$G = \text{GL}^{+}(n, \text{R})$和$G = \text{SL}(n, \text{R})$，以及它们作为仿射变换$\text{R}^{n} \rtimes G$的表示形式。我们通过将群分解为可单独处理的子群和子流形，实现了不变积分以及全局参数化。在该框架下，我们展示了如何对卷积核进行参数化，以构建对仿射变换具备等变性的模型。我们在基准仿射不变分类任务上评估了模型的鲁棒性和分布外泛化能力，其表现优于此前的方案。
英文摘要：
Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods is limited by the fact that depending on the group of interest $G$, the exponential map may not be surjective. Further limitations are encountered when $G$ is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the groups $G = \text{GL}^{+}(n, \mathbb{R})$ and $G = \text{SL}(n, \mathbb{R})$, as well as their representation as affine transformations $\mathbb{R}^{n} \rtimes G$. Invariant integration as well as a global parametrization is realized by a decomposition into subgroups and submanifolds which can be handled individually. Under this framework, 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, outperforming previous proposals.
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