# Geometry of Deep Convolutional Networks

arXiv: https://arxiv.org/abs/1905.08922 · HTML: https://mlsanity.com/paper/1905.08922/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2019.

## Built on by (most cited first, 4 of 4)

- [Origami in N dimensions: How feed-forward networks manufacture linear separability](https://mlsanity.com/paper/2203.11355/) (2022)
- [The Geometric Structure of Fully-Connected ReLU Layers](https://mlsanity.com/paper/2310.03482/) (2023)
- [Understanding Distributed Representations of Concepts in Deep Neural Networks without Supervision](https://mlsanity.com/paper/2312.17285/) (2023)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)

## Abstract (arXiv)

We give a formal procedure for computing preimages of convolutional network outputs using the dual basis defined from the set of hyperplanes associated with the layers of the network. We point out the special symmetry associated with arrangements of hyperplanes of convolutional networks that take the form of regular multidimensional polyhedral cones. We discuss the efficiency of large number of layers of nested cones that result from incremental small size convolutions in order to give a good compromise between efficient contraction of data to low dimensions and shaping of preimage manifolds. We demonstrate how a specific network flattens a non linear input manifold to an affine output manifold and discuss its relevance to understanding classification properties of deep networks.