# Orthogonal Polynomial Approximation for Matrix Log Normalization in Global Covariance Pooling

arXiv: https://arxiv.org/abs/2608.19021 · HTML: https://mlsanity.com/paper/2608.19021/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 38 references that are in the map.
- First posted on arXiv 19 Aug 2026.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [CutMix: Regularization Strategy to Train Strong Classifiers with Localizable Features](https://mlsanity.com/paper/1905.04899/) (2019)
- [RandAugment: Practical automated data augmentation with a reduced search space](https://mlsanity.com/paper/1909.13719/) (2019)
- [Fine-Grained Visual Classification of Aircraft](https://mlsanity.com/paper/1306.5151/) (2013)
- [EfficientNetV2: Smaller Models and Faster Training](https://mlsanity.com/paper/2104.00298/) (2021)
- [Towards Faster Training of Global Covariance Pooling Networks by Iterative Matrix Square Root Normalization](https://mlsanity.com/paper/1712.01034/) (2017)
- [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)
- [Covariance Pooling For Facial Expression Recognition](https://mlsanity.com/paper/1805.04855/) (2018)
- [Why Approximate Matrix Square Root Outperforms Accurate SVD in Global Covariance Pooling?](https://mlsanity.com/paper/2105.02498/) (2021)
- [Multi-Objective Matrix Normalization for Fine-grained Visual Recognition](https://mlsanity.com/paper/2003.13272/) (2020)
- [Fast Differentiable Matrix Square Root](https://mlsanity.com/paper/2201.08663/) (2022)
- [On the Eigenvalues of Global Covariance Pooling for Fine-grained Visual Recognition](https://mlsanity.com/paper/2205.13282/) (2022)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [Learning Partial Correlation based Deep Visual Representation for Image Classification](https://mlsanity.com/paper/2304.11597/) (2023)
- [SoT: Delving Deeper into Classification Head for Transformer](https://mlsanity.com/paper/2104.10935/) (2021)

## Abstract (arXiv)

Global Covariance Pooling (GCP) improves deep networks by capturing second-order feature statistics, and is especially effective for fine-grained recognition. Because covariance matrices live on the Symmetric Positive Definite (SPD) manifold, a normalization step is required before the Euclidean classifier. The faithful choice is the matrix logarithm (MLN-COV), which maps the SPD manifold to its tangent space; in practice it was abandoned in favour of the matrix square root because its eigendecomposition-based gradient is numerically unstable. We show that this instability is an artifact of computing the logarithm spectrally, not of the logarithm itself. Approximating the logarithm with finite polynomials in the covariance matrix removes the eigendecomposition from both passes: every operation becomes a General Matrix Multiplication (GEMM), the gradient stays bounded on the spectral support of the pre-normalized covariance, and the unstable 1/(lambda_i-lambda_j) term never appears. The key ingredient is a mean-eigenvalue pre-normalization that centres the spectrum near 1, away from the singularity of log, with a scalar post-compensation that returns the singular part of log(A) in closed form. Our recommended normalizer is a degree-8 Chebyshev expansion evaluated by a three-term matrix recurrence, with a matching reverse recurrence for the backward pass; Legendre, Laguerre, Taylor and Pade expansions are studied as controls that isolate the roles of the basis and of the target function. On three fine-grained benchmarks and ImageNet-1k the decomposition-free logarithm is both faster and more accurate than the spectral logarithm and than the square-root approximations it replaces, and at matched basis and degree the log target beats the square-root target, confirming that the gain comes from the faithful Riemannian map rather than from a better polynomial family.