# Notes on: Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., & Vandergheynst, P. (2017): Geometric deep learning: going beyond euclidean data

## Table of Contents

## Motivation

- One of the key reasons for success of deep learning is
*stationarity*and*composability*through local statistics - CNNs on images have
*stationarity*due to shift-invariance of an image,*locality*due to the local connectivity, and*composibility*stems from the multi-resolution structure of the grid - These properties are exploited by CNNs:
- alternating convolution and downsampling (pooling)
*convolutions*have a two-fold effect:- Extracts local features that are shared across the image domain, greatly reduces the number of parameters in the network
- Convolutional architecture itself imposes some priors about the data, which appear very suitable for images

## Geometric learning problems

Two classes of geomtric learning problems:

**Characterize the structure**of the data**Analyzing functions**defined on a given non-Euclidean domain

### Structure of the domain

- Assume to be given a set od data points of underlying lower dimensional structure embedded in a high-dimensional Euclidean space
- Recovering the lower-dimensional structure is often referred to as manifold learning

#### Manifold learning / non-linear dimensionality reduction

Many methods follow the recipe:

- Construct a representation of local affinity / similiarity of the data points (typically, sparsely connected graph)
- Data points are embedded into a low-dimensional space trying to preserve some criterion of the original affinity