Enumeration of max-pooling responses with generalized permutohedra

TL;DR

This paper explores the combinatorial structure of max-pooling layers in neural networks by linking them to generalized permutohedra, providing formulas for their geometric complexity based on pooling parameters.

Contribution

It introduces a novel geometric framework for analyzing max-pooling functions using Minkowski sums of simplices, deriving explicit formulas for their vertices and facets.

Findings

Derived formulas for the number of linear regions in 1D max-pooling.

Characterized faces of the associated polytopes.

Provided enumeration results for 2D max-pooling.

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.