# The Spectrum of Fisher Information of Deep Networks Achieving Dynamical Isometry

arXiv: https://arxiv.org/abs/2006.07814 · HTML: https://mlsanity.com/paper/2006.07814/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2020.

## Built on by (most cited first, 9 of 9)

- [Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks](https://mlsanity.com/paper/2109.09304/) (2021)
- [Asymptotic Freeness of Layerwise Jacobians Caused by Invariance of Multilayer Perceptron: The Haar Orthogonal Case](https://mlsanity.com/paper/2103.13466/) (2021)
- [Efficient Activation Function Optimization through Surrogate Modeling](https://mlsanity.com/paper/2301.05785/) (2023)
- [Gaussian Equivalence for Self-Attention: Asymptotic Spectral Analysis of Attention Matrix](https://mlsanity.com/paper/2510.06685/) (2025)
- [Dynamical Isometry for Residual Networks](https://mlsanity.com/paper/2210.02411/) (2022)
- [Free Random Projection for In-Context Reinforcement Learning](https://mlsanity.com/paper/2504.06983/) (2025)
- [Lost in Retraining: Roaming the Parameter Space of Exponential Families Under Closed-Loop Learning](https://mlsanity.com/paper/2506.20623/) (2025)
- [Fisher Information, Training and Bias in Fourier Regression Models](https://mlsanity.com/paper/2510.06945/) (2025)
- [Input-Dependent Fisher Information for Local Sensitivity Analysis of Medical Image Classifiers](https://mlsanity.com/paper/2606.16362/) (2026)

## Abstract (arXiv)

The Fisher information matrix (FIM) is fundamental to understanding the trainability of deep neural nets (DNN), since it describes the parameter space's local metric. We investigate the spectral distribution of the conditional FIM, which is the FIM given a single sample, by focusing on fully-connected networks achieving dynamical isometry. Then, while dynamical isometry is known to keep specific backpropagated signals independent of the depth, we find that the parameter space's local metric linearly depends on the depth even under the dynamical isometry. More precisely, we reveal that the conditional FIM's spectrum concentrates around the maximum and the value grows linearly as the depth increases. To examine the spectrum, considering random initialization and the wide limit, we construct an algebraic methodology based on the free probability theory. As a byproduct, we provide an analysis of the solvable spectral distribution in two-hidden-layer cases. Lastly, experimental results verify that the appropriate learning rate for the online training of DNNs is in inverse proportional to depth, which is determined by the conditional FIM's spectrum.