论文标题
PDO-ECONVS：基于部分差分运算符的模棱两可的卷积
PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions
论文作者
论文摘要
最近的研究表明，将均衡力纳入神经网络架构非常有帮助，并且有一些研究调查了小组行动下网络的均衡性。但是，由于数字图像和特征图在离散网格格里德上，因此相应的均衡转换组非常有限。在这项工作中，我们从卷积与部分差异操作员（PDOS）之间的联系中处理了这个问题。从理论上讲，假设输入是平滑的，我们会改变PDOS，并提出了一个与更通用的连续群体，即$ n $ dimension Euclidean组相等的系统。在实施中，我们使用PDOS的数值方案离散系统，从而得出了大致的卷积（PDO-ECONVS）。从理论上讲，PDO-ECONV的近似误差是二次顺序。这是当近似值时第一次提供误差分析。对旋转MNIST和自然图像分类进行的广泛实验表明，PDO-ECONVS的性能性能更有效地使用参数。特别是，与宽重的重新结合相比，我们的方法仅使用12.6％的参数产生更好的结果。
Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more general continuous group, the $n$-dimension Euclidean group. In implementation, we discretize the system using the numerical schemes of PDOs, deriving approximately equivariant convolutions (PDO-eConvs). Theoretically, the approximation error of PDO-eConvs is of the quadratic order. It is the first time that the error analysis is provided when the equivariance is approximate. Extensive experiments on rotated MNIST and natural image classification show that PDO-eConvs perform competitively yet use parameters much more efficiently. Particularly, compared with Wide ResNets, our methods result in better results using only 12.6% parameters.
