Enumeration of max-pooling responses with generalized permutohedra
Abstract
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.
Keywords
Cite
@article{arxiv.2209.14978,
title = {Enumeration of max-pooling responses with generalized permutohedra},
author = {Laura Escobar and Patricio Gallardo and Javier González-Anaya and José L. González and Guido Montúfar and Alejandro H. Morales},
journal= {arXiv preprint arXiv:2209.14978},
year = {2023}
}
Comments
35 pages, 11 figures, 4 tables. V2: Improved exposition, added computations in Section 4, and expanded analysis of data