# Linearly Constrained Weights: Reducing Activation Shift for Faster Training of Neural Networks

arXiv: https://arxiv.org/abs/2403.13833 · HTML: https://mlsanity.com/paper/2403.13833/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 24 references that are in the map.
- First posted on arXiv 8 Mar 2024.

## Builds on (most cited first, 13 of 13)

- [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)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [Spectral Norm Regularization for Improving the Generalizability of Deep Learning](https://mlsanity.com/paper/1705.10941/) (2017)
- [Knowledge Matters: Importance of Prior Information for Optimization](https://mlsanity.com/paper/1301.4083/) (2013)

## Built on by (most cited first, 2 of 2)

- [Hidden Boundary Motion in Transformer Optimization: Function-Space Orthogonalization of Affine Weight and Bias Updates](https://mlsanity.com/paper/2607.22927/) (2026)
- [Rethinking Activation Functions: Reducing Bias Can Improve Learning](https://mlsanity.com/paper/2610.14500/) (2026)

## Abstract (arXiv)

In this paper, we first identify activation shift, a simple but remarkable phenomenon in a neural network in which the preactivation value of a neuron has non-zero mean that depends on the angle between the weight vector of the neuron and the mean of the activation vector in the previous layer. We then propose linearly constrained weights (LCW) to reduce the activation shift in both fully connected and convolutional layers. The impact of reducing the activation shift in a neural network is studied from the perspective of how the variance of variables in the network changes through layer operations in both forward and backward chains. We also discuss its relationship to the vanishing gradient problem. Experimental results show that LCW enables a deep feedforward network with sigmoid activation functions to be trained efficiently by resolving the vanishing gradient problem. Moreover, combined with batch normalization, LCW improves generalization performance of both feedforward and convolutional networks.