# Fixed points of nonnegative neural networks

arXiv: https://arxiv.org/abs/2106.16239 · HTML: https://mlsanity.com/paper/2106.16239/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 81 references that are in the map.
- First posted on arXiv 30 Jun 2021.

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

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Learning to Detect](https://mlsanity.com/paper/1805.07631/) (2018)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators](https://mlsanity.com/paper/1903.01014/) (2019)
- [Recurrent Inference Machines for Solving Inverse Problems](https://mlsanity.com/paper/1706.04008/) (2017)
- [Adaptive Neural Signal Detection for Massive MIMO](https://mlsanity.com/paper/1906.04610/) (2019)
- [Deep Neural Network Structures Solving Variational Inequalities](https://mlsanity.com/paper/1808.07526/) (2018)
- [Gamma Belief Networks](https://mlsanity.com/paper/1512.03081/) (2015)
- [Deep Learning of Nonnegativity-Constrained Autoencoders for Enhanced Understanding of Data](https://mlsanity.com/paper/1802.00003/) (2018)
- [Deep Learning of Part-based Representation of Data Using Sparse Autoencoders with Nonnegativity Constraints](https://mlsanity.com/paper/1601.02733/) (2016)
- [Connections between spectral properties of asymptotic mappings and solutions to wireless network problems](https://mlsanity.com/paper/1903.09793/) (2019)
- [Elementary Properties of Positive Concave Mappings with Applications to Network Planning and Optimization](https://mlsanity.com/paper/1505.03006/) (2015)
- [Toward Energy-Efficient 5G Wireless Communications Technologies](https://mlsanity.com/paper/1407.0344/) (2014)
- [Spectral radii of asymptotic mappings and the convergence speed of the standard fixed point algorithm](https://mlsanity.com/paper/1803.05671/) (2018)
- [Iterative Neural Networks with Bounded Weights](https://mlsanity.com/paper/1908.05982/) (2019)

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

- [LAMBDA: A Large Model Based Data Agent](https://mlsanity.com/paper/2407.17535/) (2024)
- [Positive concave deep equilibrium models](https://mlsanity.com/paper/2402.04029/) (2024)
- [Are We Measuring Oversmoothing in Graph Neural Networks Correctly?](https://mlsanity.com/paper/2502.04591/) (2025)
- [Subhomogeneous Deep Equilibrium Models](https://mlsanity.com/paper/2403.00720/) (2024)
- [Random weights of DNNs and emergence of fixed points](https://mlsanity.com/paper/2501.04182/) (2025)

## Abstract (arXiv)

We use fixed point theory to analyze nonnegative neural networks, which we define as neural networks that map nonnegative vectors to nonnegative vectors. We first show that nonnegative neural networks with nonnegative weights and biases can be recognized as monotonic and (weakly) scalable mappings within the framework of nonlinear Perron-Frobenius theory. This fact enables us to provide conditions for the existence of fixed points of nonnegative neural networks having inputs and outputs of the same dimension, and these conditions are weaker than those recently obtained using arguments in convex analysis. Furthermore, we prove that the shape of the fixed point set of nonnegative neural networks with nonnegative weights and biases is an interval, which under mild conditions degenerates to a point. These results are then used to obtain the existence of fixed points of more general nonnegative neural networks. From a practical perspective, our results contribute to the understanding of the behavior of autoencoders, and we also offer valuable mathematical machinery for future developments in deep equilibrium models.