This work presents a novel algorithm for transforming a neural network into a spline representation. Unlike previous work that required convex and piecewise-affine network operators to create a max-affine spline alternate form, this work relaxes this constraint. The only constraint is that the function be bounded and possess a well-define second derivative, although this was shown experimentally to not be strictly necessary. It can also be performed over the whole network rather than on each layer independently. As in previous work, this bridges the gap between neural networks and approximation theory but also enables the visualisation of network feature maps. Mathematical proof and experimental investigation of the technique is performed with approximation error and feature maps being extracted from a range of architectures, including convolutional neural networks.

本论文提出了一种将神经网络转化为样条表示的新算法，通过放宽先前工作对凸和分段仿射网络算子的要求，该算法仅需要函数被约束为有界且具有良好定义的二阶导数，实验证明这并非必需。该算法可适用于整个网络，而不是独立于每一层进行操作，从而桥接了神经网络和逼近理论之间的差距，并使得网络特征图可视化成为可能，通过近似误差和特征图的提取，对一系列架构包括卷积神经网络进行了数学证明和实验研究。

使用凸凹表示的Legendre变换对神经网络进行最大仿射样条逼近