# Transition to Linearity of General Neural Networks with Directed Acyclic Graph Architecture

arXiv: https://arxiv.org/abs/2205.11786 · HTML: https://mlsanity.com/paper/2205.11786/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 46 references that are in the map.
- First posted on arXiv 24 May 2022.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [An Improved Analysis of Training Over-parameterized Deep Neural Networks](https://mlsanity.com/paper/1906.04688/) (2019)
- [Infinite attention: NNGP and NTK for deep attention networks](https://mlsanity.com/paper/2006.10540/) (2020)
- [Finite Depth and Width Corrections to the Neural Tangent Kernel](https://mlsanity.com/paper/1909.05989/) (2019)
- [Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks](https://mlsanity.com/paper/2012.11654/) (2020)
- [The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training](https://mlsanity.com/paper/2007.12826/) (2020)
- [Graph Structure of Neural Networks](https://mlsanity.com/paper/2007.06559/) (2020)
- [The Recurrent Neural Tangent Kernel](https://mlsanity.com/paper/2006.10246/) (2020)
- [Discovering Neural Wirings](https://mlsanity.com/paper/1906.00586/) (2019)
- [Transition to Linearity of Wide Neural Networks is an Emerging Property of Assembling Weak Models](https://mlsanity.com/paper/2203.05104/) (2022)

## Built on by (most cited first, 4 of 4)

- [Catapults in SGD: spikes in the training loss and their impact on generalization through feature learning](https://mlsanity.com/paper/2306.04815/) (2023)
- [Toward Large Kernel Models](https://mlsanity.com/paper/2302.02605/) (2023)
- [On Emergence of Clean-Priority Learning in Early Stopped Neural Networks](https://mlsanity.com/paper/2306.02533/) (2023)
- [Label-NTK Alignments and A Tighter Convergence Bound in the NTK Regime](https://mlsanity.com/paper/2605.25275/) (2026)

## Abstract (arXiv)

In this paper we show that feedforward neural networks corresponding to arbitrary directed acyclic graphs undergo transition to linearity as their "width" approaches infinity. The width of these general networks is characterized by the minimum in-degree of their neurons, except for the input and first layers. Our results identify the mathematical structure underlying transition to linearity and generalize a number of recent works aimed at characterizing transition to linearity or constancy of the Neural Tangent Kernel for standard architectures.