# Training Sparse Neural Network by Constraining Synaptic Weight on Unit Lp Sphere

arXiv: https://arxiv.org/abs/2103.16013 · HTML: https://mlsanity.com/paper/2103.16013/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 34 references that are in the map.
- First posted on arXiv 30 Mar 2021.

## Builds on (most cited first, 18 of 18)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [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)
- [Learning Deep Features for Discriminative Localization](https://mlsanity.com/paper/1512.04150/) (2015)
- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [Rigging the Lottery: Making All Tickets Winners](https://mlsanity.com/paper/1911.11134/) (2019)
- [The State of Sparsity in Deep Neural Networks](https://mlsanity.com/paper/1902.09574/) (2019)
- [Scalable Training of Artificial Neural Networks with Adaptive Sparse Connectivity inspired by Network Science](https://mlsanity.com/paper/1707.04780/) (2017)
- [Efficient Neural Audio Synthesis](https://mlsanity.com/paper/1802.08435/) (2018)
- [Fast Sparse ConvNets](https://mlsanity.com/paper/1911.09723/) (2019)
- [Exploring Sparsity in Recurrent Neural Networks](https://mlsanity.com/paper/1704.05119/) (2017)
- [Training Sparse Neural Networks](https://mlsanity.com/paper/1611.06694/) (2016)
- [Automatic Knee Osteoarthritis Diagnosis from Plain Radiographs: A Deep Learning-Based Approach](https://mlsanity.com/paper/1710.10589/) (2017)
- [Learning Sparse Neural Networks via Sensitivity-Driven Regularization](https://mlsanity.com/paper/1810.11764/) (2018)
- [Understanding brain networks and brain organization](https://mlsanity.com/paper/1403.7151/) (2014)
- [Sparseout: Controlling Sparsity in Deep Networks](https://mlsanity.com/paper/1904.08050/) (2019)

## Abstract (arXiv)

Sparse deep neural networks have shown their advantages over dense models with fewer parameters and higher computational efficiency. Here we demonstrate constraining the synaptic weights on unit Lp-sphere enables the flexibly control of the sparsity with p and improves the generalization ability of neural networks. Firstly, to optimize the synaptic weights constrained on unit Lp-sphere, the parameter optimization algorithm, Lp-spherical gradient descent (LpSGD) is derived from the augmented Empirical Risk Minimization condition, which is theoretically proved to be convergent. To understand the mechanism of how p affects Hoyer's sparsity, the expectation of Hoyer's sparsity under the hypothesis of gamma distribution is given and the predictions are verified at various p under different conditions. In addition, the "semi-pruning" and threshold adaptation are designed for topology evolution to effectively screen out important connections and lead the neural networks converge from the initial sparsity to the expected sparsity. Our approach is validated by experiments on benchmark datasets covering a wide range of domains. And the theoretical analysis pave the way to future works on training sparse neural networks with constrained optimization.