# Neural Characteristic Activation Analysis and Geometric Parameterization for ReLU Networks

arXiv: https://arxiv.org/abs/2305.15912 · HTML: https://mlsanity.com/paper/2305.15912/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 71 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 30 of 50)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Instance Normalization: The Missing Ingredient for Fast Stylization](https://mlsanity.com/paper/1607.08022/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (2017)
- [Limitations of the Empirical Fisher Approximation for Natural Gradient Descent](https://mlsanity.com/paper/1905.12558/) (2019)
- [Fixup Initialization: Residual Learning Without Normalization](https://mlsanity.com/paper/1901.09321/) (2019)
- [Client Selection in Federated Learning: Convergence Analysis and Power-of-Choice Selection Strategies](https://mlsanity.com/paper/2010.01243/) (2020)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [L2 Regularization versus Batch and Weight Normalization](https://mlsanity.com/paper/1706.05350/) (2017)
- [Optimal Client Sampling for Federated Learning](https://mlsanity.com/paper/2010.13723/) (2020)
- [An Exponential Learning Rate Schedule for Deep Learning](https://mlsanity.com/paper/1910.07454/) (2019)

## Built on by (most cited first, 5 of 5)

- [Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training](https://mlsanity.com/paper/2311.18022/) (2023)
- [Structured Transformations for Stable and Interpretable Neural Computation](https://mlsanity.com/paper/2508.00127/) (2025)
- [Capacity Matters: a Proof-of-Concept for Transformer Memorization on Real-World Data](https://mlsanity.com/paper/2506.14704/) (2025)
- [A Composite Activation Function for Learning Stable Binary Representations](https://mlsanity.com/paper/2605.11558/) (2026)
- [Affine Geometry of Gaussian ReLU Networks via Conditional Kac-Rice Formulas](https://mlsanity.com/paper/2609.32695/) (2026)

## Abstract (arXiv)

We introduce a novel approach for analyzing the training dynamics of ReLU networks by examining the characteristic activation boundaries of individual ReLU neurons. Our proposed analysis reveals a critical instability in common neural network parameterizations and normalizations during stochastic optimization, which impedes fast convergence and hurts generalization performance. Addressing this, we propose Geometric Parameterization (GmP), a novel neural network parameterization technique that effectively separates the radial and angular components of weights in the hyperspherical coordinate system. We show theoretically that GmP resolves the aforementioned instability issue. We report empirical results on various models and benchmarks to verify GmP's advantages of optimization stability, convergence speed and generalization performance.