# Emergence of Globally Attracting Fixed Points in Deep Neural Networks With Nonlinear Activations

arXiv: https://arxiv.org/abs/2410.20107 · HTML: https://mlsanity.com/paper/2410.20107/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 27 references that are in the map.
- First posted on arXiv 26 Oct 2024.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Deep Information Propagation](https://mlsanity.com/paper/1611.01232/) (2016)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation](https://mlsanity.com/paper/1902.04760/) (2019)
- [Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse](https://mlsanity.com/paper/2206.03126/) (2022)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice](https://mlsanity.com/paper/1711.04735/) (2017)
- [On the Impact of the Activation Function on Deep Neural Networks Training](https://mlsanity.com/paper/1902.06853/) (2019)
- [The Emergence of Spectral Universality in Deep Networks](https://mlsanity.com/paper/1802.09979/) (2018)
- [A Mean Field Theory of Batch Normalization](https://mlsanity.com/paper/1902.08129/) (2019)
- [The Neural Covariance SDE: Shaped Infinite Depth-and-Width Networks at Initialization](https://mlsanity.com/paper/2206.02768/) (2022)
- [Batch Normalization Orthogonalizes Representations in Deep Random Networks](https://mlsanity.com/paper/2106.03970/) (2021)
- [On the impact of activation and normalization in obtaining isometric embeddings at initialization](https://mlsanity.com/paper/2305.18399/) (2023)
- [Towards Training Without Depth Limits: Batch Normalization Without Gradient Explosion](https://mlsanity.com/paper/2310.02012/) (2023)
- [On Bridging the Gap between Mean Field and Finite Width in Deep Random Neural Networks with Batch Normalization](https://mlsanity.com/paper/2205.13076/) (2022)

## Built on by (most cited first, 1 of 1)

- [Barriers for Learning in an Evolving World: Mathematical Understanding of Loss of Plasticity](https://mlsanity.com/paper/2510.00304/) (2025)

## Abstract (arXiv)

Understanding how neural networks transform input data across layers is fundamental to unraveling their learning and generalization capabilities. Although prior work has used insights from kernel methods to study neural networks, a global analysis of how the similarity between hidden representations evolves across layers remains underexplored. In this paper, we introduce a theoretical framework for the evolution of the kernel sequence, which measures the similarity between the hidden representation for two different inputs. Operating under the mean-field regime, we show that the kernel sequence evolves deterministically via a kernel map, which only depends on the activation function. By expanding activation using Hermite polynomials and using their algebraic properties, we derive an explicit form for kernel map and fully characterize its fixed points. Our analysis reveals that for nonlinear activations, the kernel sequence converges globally to a unique fixed point, which can correspond to orthogonal or similar representations depending on the activation and network architecture. We further extend our results to networks with residual connections and normalization layers, demonstrating similar convergence behaviors. This work provides new insights into the implicit biases of deep neural networks and how architectural choices influence the evolution of representations across layers.