# From Compass and Ruler to Convolution and Nonlinearity: On the Surprising Difficulty of Understanding a Simple CNN Solving a Simple Geometric Estimation Task

arXiv: https://arxiv.org/abs/2303.06638 · HTML: https://mlsanity.com/paper/2303.06638/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 15 references that are in the map.
- First posted on arXiv 12 Mar 2023.

## Builds on (most cited first, 9 of 9)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [European Union regulations on algorithmic decision-making and a "right to explanation"](https://mlsanity.com/paper/1606.08813/) (2016)
- [On Interpretability of Artificial Neural Networks: A Survey](https://mlsanity.com/paper/2001.02522/) (2020)

## Built on by (most cited first, 2 of 2)

- [Metric Convolutions: A Unifying Theory to Adaptive Image Convolutions](https://mlsanity.com/paper/2406.05400/) (2024)
- [A model is worth tens of thousands of examples](https://mlsanity.com/paper/2303.10608/) (2023)

## Abstract (arXiv)

Neural networks are omnipresent, but remain poorly understood. Their increasing complexity and use in critical systems raises the important challenge to full interpretability. We propose to address a simple well-posed learning problem: estimating the radius of a centred pulse in a one-dimensional signal or of a centred disk in two-dimensional images using a simple convolutional neural network. Surprisingly, understanding what trained networks have learned is difficult and, to some extent, counter-intuitive. However, an in-depth theoretical analysis in the one-dimensional case allows us to comprehend constraints due to the chosen architecture, the role of each filter and of the nonlinear activation function, and every single value taken by the weights of the model. Two fundamental concepts of neural networks arise: the importance of invariance and of the shape of the nonlinear activation functions.