# Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training

arXiv: https://arxiv.org/abs/2311.18022 · HTML: https://mlsanity.com/paper/2311.18022/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 42 references that are in the map.
- First posted on arXiv 29 Nov 2023.

## Builds on (most cited first, 24 of 24)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations](https://mlsanity.com/paper/1708.02691/) (2017)
- [Representation Benefits of Deep Feedforward Networks](https://mlsanity.com/paper/1509.08101/) (2015)
- [Reverse-Engineering Deep ReLU Networks](https://mlsanity.com/paper/1910.00744/) (2019)
- [Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions](https://mlsanity.com/paper/1807.00297/) (2018)
- [A comparison of deep networks with ReLU activation function and linear spline-type methods](https://mlsanity.com/paper/1804.02253/) (2018)
- [How degenerate is the parametrization of neural networks with the ReLU activation function?](https://mlsanity.com/paper/1905.09803/) (2019)
- [The universal approximation power of finite-width deep ReLU networks](https://mlsanity.com/paper/1806.01528/) (2018)
- [SERF: Towards better training of deep neural networks using log-Softplus ERror activation Function](https://mlsanity.com/paper/2108.09598/) (2021)
- [Neural Characteristic Activation Analysis and Geometric Parameterization for ReLU Networks](https://mlsanity.com/paper/2305.15912/) (2023)
- [Unsupervised Representation Learning via Neural Activation Coding](https://mlsanity.com/paper/2112.04014/) (2021)
- [Initializing ReLU networks in an expressive subspace of weights](https://mlsanity.com/paper/2103.12499/) (2021)

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

- [Training-Time Batch Normalization Reshapes Local Partition Geometry in Piecewise-Affine Networks](https://mlsanity.com/paper/2605.04946/) (2026)
- [AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networks](https://mlsanity.com/paper/2605.06218/) (2026)
- [Region Seeding via Pre-Activation Regularization: A Geometric View of Piecewise Affine Neural Networks](https://mlsanity.com/paper/2605.06300/) (2026)
- [Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexity](https://mlsanity.com/paper/2606.21687/) (2026)

## Abstract (arXiv)

In a neural network with ReLU activations, the number of piecewise linear regions in the output can grow exponentially with depth. However, this is highly unlikely to happen when the initial parameters are sampled randomly, which therefore often leads to the use of networks that are unnecessarily large. To address this problem, we introduce a novel parameterization of the network that restricts its weights so that a depth d network produces exactly 2d linear regions at initialization and maintains those regions throughout training under the parameterization. This approach allows us to learn approximations of convex, one dimensional functions that are several orders of magnitude more accurate than their randomly initialized counterparts. We further demonstrate a preliminary extension of our construction to multidimensional and non-convex functions, allowing the technique to replace traditional dense layers in various architectures.