# DeepKSPD: Learning Kernel-matrix-based SPD Representation for Fine-grained Image Recognition

arXiv: https://arxiv.org/abs/1711.04047 · HTML: https://mlsanity.com/paper/1711.04047/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Nov 2017.

## Built on by (most cited first, 7 of 7)

- [Fine-Grained Image Analysis with Deep Learning: A Survey](https://mlsanity.com/paper/2111.06119/) (2021)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [Learning Partial Correlation based Deep Visual Representation for Image Classification](https://mlsanity.com/paper/2304.11597/) (2023)
- [Scalable Deep Unfolding of Conic Optimizers](https://mlsanity.com/paper/2606.13825/) (2026)
- [DALIP: Distribution Alignment-based Language-Image Pre-Training for Domain-Specific Data](https://mlsanity.com/paper/2504.01386/) (2025)
- [Deep and Shallow Covariance Feature Quantization for 3D Facial Expression Recognition](https://mlsanity.com/paper/2105.05708/) (2021)
- [A Deeper Look into Second-Order Feature Aggregation for LiDAR Place Recognition](https://mlsanity.com/paper/2409.15919/) (2024)

## Abstract (arXiv)

Being symmetric positive-definite (SPD), covariance matrix has traditionally been used to represent a set of local descriptors in visual recognition. Recent study shows that kernel matrix can give considerably better representation by modelling the nonlinearity in the local descriptor set. Nevertheless, neither the descriptors nor the kernel matrix is deeply learned. Worse, they are considered separately, hindering the pursuit of an optimal SPD representation. This work proposes a deep network that jointly learns local descriptors, kernel-matrix-based SPD representation, and the classifier via an end-to-end training process. We derive the derivatives for the mapping from a local descriptor set to the SPD representation to carry out backpropagation. Also, we exploit the Daleckii-Krein formula in operator theory to give a concise and unified result on differentiating SPD matrix functions, including the matrix logarithm to handle the Riemannian geometry of kernel matrix. Experiments not only show the superiority of kernel-matrix-based SPD representation with deep local descriptors, but also verify the advantage of the proposed deep network in pursuing better SPD representations for fine-grained image recognition tasks.