# Fast Differentiable Matrix Square Root

arXiv: https://arxiv.org/abs/2201.08663 · HTML: https://mlsanity.com/paper/2201.08663/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 45 references that are in the map.
- First posted on arXiv 21 Jan 2022.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Universal Style Transfer via Feature Transforms](https://mlsanity.com/paper/1705.08086/) (2017)
- [Whitening for Self-Supervised Representation Learning](https://mlsanity.com/paper/2007.06346/) (2020)
- [Bilinear CNNs for Fine-grained Visual Recognition](https://mlsanity.com/paper/1504.07889/) (2015)
- [Improved Texture Networks: Maximizing Quality and Diversity in Feed-forward Stylization and Texture Synthesis](https://mlsanity.com/paper/1701.02096/) (2017)
- [RobustNet: Improving Domain Generalization in Urban-Scene Segmentation via Instance Selective Whitening](https://mlsanity.com/paper/2103.15597/) (2021)
- [Switchable Whitening for Deep Representation Learning](https://mlsanity.com/paper/1904.09739/) (2019)
- [Iterative Normalization: Beyond Standardization towards Efficient Whitening](https://mlsanity.com/paper/1904.03441/) (2019)
- [Decorrelated Batch Normalization](https://mlsanity.com/paper/1804.08450/) (2018)
- [Towards Faster Training of Global Covariance Pooling Networks by Iterative Matrix Square Root Normalization](https://mlsanity.com/paper/1712.01034/) (2017)
- [Image-to-Image Translation via Group-wise Deep Whitening-and-Coloring Transformation](https://mlsanity.com/paper/1812.09912/) (2018)
- [Is Second-order Information Helpful for Large-scale Visual Recognition?](https://mlsanity.com/paper/1703.08050/) (2017)
- [Deep CNNs Meet Global Covariance Pooling: Better Representation and Generalization](https://mlsanity.com/paper/1904.06836/) (2019)
- [Diversified Arbitrary Style Transfer via Deep Feature Perturbation](https://mlsanity.com/paper/1909.08223/) (2019)
- [Training Deep Networks with Structured Layers by Matrix Backpropagation](https://mlsanity.com/paper/1509.07838/) (2015)
- [Improved Bilinear Pooling with CNNs](https://mlsanity.com/paper/1707.06772/) (2017)
- [Whitening and Coloring batch transform for GANs](https://mlsanity.com/paper/1806.00420/) (2018)
- [Backpropagation-Friendly Eigendecomposition](https://mlsanity.com/paper/1906.09023/) (2019)
- [Why Approximate Matrix Square Root Outperforms Accurate SVD in Global Covariance Pooling?](https://mlsanity.com/paper/2105.02498/) (2021)
- [Robust Differentiable SVD](https://mlsanity.com/paper/2104.03821/) (2021)
- [Keep it Simple: Image Statistics Matching for Domain Adaptation](https://mlsanity.com/paper/2005.12551/) (2020)
- [Eigendecomposition-free Training of Deep Networks with Zero Eigenvalue-based Losses](https://mlsanity.com/paper/1803.08071/) (2018)
- [An Investigation into the Stochasticity of Batch Whitening](https://mlsanity.com/paper/2003.12327/) (2020)
- [Group Whitening: Balancing Learning Efficiency and Representational Capacity](https://mlsanity.com/paper/2009.13333/) (2020)
- [Eigendecomposition-Free Training of Deep Networks for Linear Least-Square Problems](https://mlsanity.com/paper/2004.07931/) (2020)

## Built on by (most cited first, 14 of 14)

- [Fast Differentiable Matrix Square Root and Inverse Square Root](https://mlsanity.com/paper/2201.12543/) (2022)
- [On the Eigenvalues of Global Covariance Pooling for Fine-grained Visual Recognition](https://mlsanity.com/paper/2205.13282/) (2022)
- [Enhancing LLM Training via Spectral Clipping](https://mlsanity.com/paper/2603.14315/) (2026)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [Grouping-matrix based Graph Pooling with Adaptive Number of Clusters](https://mlsanity.com/paper/2209.02939/) (2022)
- [Orthogonal SVD Covariance Conditioning and Latent Disentanglement](https://mlsanity.com/paper/2212.05599/) (2022)
- [Improving Covariance Conditioning of the SVD Meta-layer by Orthogonality](https://mlsanity.com/paper/2207.02119/) (2022)
- [MuonSSM: Orthogonalizing State Space Models for Sequence Modeling](https://mlsanity.com/paper/2606.30461/) (2026)
- [Batch-efficient EigenDecomposition for Small and Medium Matrices](https://mlsanity.com/paper/2207.04228/) (2022)
- [Series of Hessian-Vector Products for Tractable Saddle-Free Newton Optimisation of Neural Networks](https://mlsanity.com/paper/2310.14901/) (2023)
- [MatRL: Provably Generalizable Iterative Algorithm Discovery via Monte-Carlo Tree Search](https://mlsanity.com/paper/2507.03833/) (2025)
- [Multivariate Functional Linear Discriminant Analysis for the Classification of Short Time Series with Missing Data](https://mlsanity.com/paper/2402.13103/) (2024)
- [Orthogonal Polynomial Approximation for Matrix Log Normalization in Global Covariance Pooling](https://mlsanity.com/paper/2608.19021/) (2026)
- [Fast Differentiable SVD on GPU via Polar Decomposition](https://mlsanity.com/paper/2609.32505/) (2026)

## Abstract (arXiv)

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at hrefhttps://github.com/KingJamesSong/FastDifferentiableMatSqrthttps://github.com/KingJamesSong/FastDifferentiableMatSqrt.