# Learning a Robust Representation via a Deep Network on Symmetric Positive Definite Manifolds

arXiv: https://arxiv.org/abs/1711.06540 · HTML: https://mlsanity.com/paper/1711.06540/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2017.

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

- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis](https://mlsanity.com/paper/2203.01631/) (2022)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)
- [Journey into Automation: Image-Derived Pavement Texture Extraction and Evaluation](https://mlsanity.com/paper/2501.02414/) (2025)
- [A Consolidated Volatility Prediction with Back Propagation Neural Network and Genetic Algorithm](https://mlsanity.com/paper/2412.07223/) (2024)
- [A Pseudo Global Fusion Paradigm-Based Cross-View Network for LiDAR-Based Place Recognition](https://mlsanity.com/paper/2508.08917/) (2025)

## Abstract (arXiv)

Recent studies have shown that aggregating convolutional features of a pre-trained Convolutional Neural Network (CNN) can obtain impressive performance for a variety of visual tasks. The symmetric Positive Definite (SPD) matrix becomes a powerful tool due to its remarkable ability to learn an appropriate statistic representation to characterize the underlying structure of visual features. In this paper, we propose to aggregate deep convolutional features into an SPD matrix representation through the SPD generation and the SPD transformation under an end-to-end deep network. To this end, several new layers are introduced in our network, including a nonlinear kernel aggregation layer, an SPD matrix transformation layer, and a vectorization layer. The nonlinear kernel aggregation layer is employed to aggregate the convolutional features into a real SPD matrix directly. The SPD matrix transformation layer is designed to construct a more compact and discriminative SPD representation. The vectorization and normalization operations are performed in the vectorization layer for reducing the redundancy and accelerating the convergence. The SPD matrix in our network can be considered as a mid-level representation bridging convolutional features and high-level semantic features. To demonstrate the effectiveness of our method, we conduct extensive experiments on visual classification. Experiment results show that our method notably outperforms state-of-the-art methods.