We investigate the combinatorics of max-pooling layers, which are functions that downsample input arrays by taking the maximum over shifted windows of input coordinates, and which are commonly used in convolutional neural networks. We obtain results on the number of linearity regions of these functions by equivalently counting the number of vertices of certain Minkowski sums of simplices. We characterize the faces of such polytopes and obtain generating functions and closed formulas for the number of vertices and facets in a 1D max-pooling layer depending on the size of the pooling windows and stride, and for the number of vertices in a special case of 2D max-pooling.

翻译：我们研究最大池化层的组合结构，这类函数通过取输入坐标滑动窗口内的最大值来对输入数组进行下采样，在卷积神经网络中应用广泛。通过等价地计算特定单形闵可夫斯基和的顶点数量，我们获得了这些函数线性区域数量的结果。我们刻画了此类多面体的面结构，推导了基于池化窗口大小和步长的一维最大池化层顶点与面数的生成函数及闭式表达式，并给出了二维最大池化层特例下的顶点数计算公式。