# Information Geometry of Orthogonal Initializations and Training

arXiv: https://arxiv.org/abs/1810.03785 · HTML: https://mlsanity.com/paper/1810.03785/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2018.

## Built on by (most cited first, 7 of 7)

- [Persistent learning signals and working memory without continuous attractors](https://mlsanity.com/paper/2308.12585/) (2023)
- [Asymptotic Freeness of Layerwise Jacobians Caused by Invariance of Multilayer Perceptron: The Haar Orthogonal Case](https://mlsanity.com/paper/2103.13466/) (2021)
- [Gradient Flossing: Improving Gradient Descent through Dynamic Control of Jacobians](https://mlsanity.com/paper/2312.17306/) (2023)
- [On the validity of kernel approximations for orthogonally-initialized neural networks](https://mlsanity.com/paper/2104.05878/) (2021)
- [MEIM: Multi-partition Embedding Interaction Beyond Block Term Format for Efficient and Expressive Link Prediction](https://mlsanity.com/paper/2209.15597/) (2022)
- [Towards the Spectral bias Alleviation by Normalizations in Coordinate Networks](https://mlsanity.com/paper/2407.17834/) (2024)
- [Information Geometry of Evolution of Neural Network Parameters While Training](https://mlsanity.com/paper/2406.05295/) (2024)

## Abstract (arXiv)

Recently mean field theory has been successfully used to analyze properties of wide, random neural networks. It gave rise to a prescriptive theory for initializing feed-forward neural networks with orthogonal weights, which ensures that both the forward propagated activations and the backpropagated gradients are near ℓ2 isometries and as a consequence training is orders of magnitude faster. Despite strong empirical performance, the mechanisms by which critical initializations confer an advantage in the optimization of deep neural networks are poorly understood. Here we show a novel connection between the maximum curvature of the optimization landscape (gradient smoothness) as measured by the Fisher information matrix (FIM) and the spectral radius of the input-output Jacobian, which partially explains why more isometric networks can train much faster. Furthermore, given that orthogonal weights are necessary to ensure that gradient norms are approximately preserved at initialization, we experimentally investigate the benefits of maintaining orthogonality throughout training, from which we conclude that manifold optimization of weights performs well regardless of the smoothness of the gradients. Moreover, motivated by experimental results we show that a low condition number of the FIM is not predictive of faster learning.