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Enumeration of max-pooling responses with generalized permutohedra

arXiv 2209.14978 ↗First posted 29 Sep 2022math.CO cs.DM cs.LGcited by 0

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Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 15 of its 44 references that are in the map. First posted on arXiv 29 Sep 2022.

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15 of 44 references.

References · 15

  1. 2014Going Deeper with Convolutionscited by 4,231
  2. 2014On the Number of Linear Regions of Deep Neural Networkscited by 378
  3. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 253
  4. 2016Benefits of depth in neural networkscited by 248
  5. 2021Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networkscited by 219
  6. 2017Bounding and Counting Linear Regions of Deep Neural Networkscited by 161
  7. 2013On the number of response regions of deep feed forward networks with piece-wise linear activationscited by 94
  8. 2018Tropical Geometry of Deep Neural Networkscited by 92
  9. 2021Towards Lower Bounds on the Depth of ReLU Neural Networkscited by 55
  10. 2019The Geometry of Deep Networks: Power Diagram Subdivisioncited by 53
  11. 2020On the Number of Linear Regions of Convolutional Neural Networkscited by 49
  12. 2021Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sumscited by 37
3 more
  1. 2018A Tropical Approach to Neural Networks with Piecewise Linear Activationscited by 26
  2. 2018A Framework for the construction of upper bounds on the number of affine linear regions of ReLU feed-forward neural networkscited by 15
  3. 2015A geometric approach for the upper bound theorem for Minkowski sums of convex polytopescited by 1

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