# Collaborative Representation for SPD Matrices with Application to Image-Set Classification

arXiv: https://arxiv.org/abs/2201.08962 · HTML: https://mlsanity.com/paper/2201.08962/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 56 references that are in the map.
- First posted on arXiv 22 Jan 2022.

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

- [NestFuse: An Infrared and Visible Image Fusion Architecture based on Nest Connection and Spatial/Channel Attention Models](https://mlsanity.com/paper/2007.00328/) (2020)
- [Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods](https://mlsanity.com/paper/1605.06182/) (2016)
- [From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices](https://mlsanity.com/paper/1407.1120/) (2014)
- [Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1412.4172/) (2014)
- [Multi-focus Image Fusion using dictionary learning and Low-Rank Representation](https://mlsanity.com/paper/1804.08355/) (2018)
- [Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach](https://mlsanity.com/paper/1304.4344/) (2013)
- [Face Detection, Bounding Box Aggregation and Pose Estimation for Robust Facial Landmark Localisation in the Wild](https://mlsanity.com/paper/1705.02402/) (2017)
- [Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences](https://mlsanity.com/paper/1409.0083/) (2014)
- [Multiple Riemannian Manifold-valued Descriptors based Image Set Classification with Multi-Kernel Metric Learning](https://mlsanity.com/paper/1908.01950/) (2019)
- [Image Set based Collaborative Representation for Face Recognition](https://mlsanity.com/paper/1308.6687/) (2013)
- [Riemannian kernel based Nystr"om method for approximate infinite-dimensional covariance descriptors with application to image set classification](https://mlsanity.com/paper/1806.06177/) (2018)
- [Multiple Manifolds Metric Learning with Application to Image Set Classification](https://mlsanity.com/paper/1805.11918/) (2018)
- [Component SPD Matrices: A lower-dimensional discriminative data descriptor for image set classification](https://mlsanity.com/paper/1806.06178/) (2018)

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

- [Multi-manifold Attention for Vision Transformers](https://mlsanity.com/paper/2207.08569/) (2022)
- [Doubly robust estimation of causal effects for random object outcomes with continuous treatments](https://mlsanity.com/paper/2506.22754/) (2025)

## Abstract (arXiv)

Collaborative representation-based classification (CRC) has demonstrated remarkable progress in the past few years because of its closed-form analytical solutions. However, the existing CRC methods are incapable of processing the nonlinear variational information directly. Recent advances illustrate that how to effectively model these nonlinear variational information and learn invariant representations is an open challenge in the community of computer vision and pattern recognition To this end, we try to design a new algorithm to handle this problem. Firstly, the second-order statistic, i.e., covariance matrix is applied to model the original image sets. Due to the space formed by a set of nonsingular covariance matrices is a well-known Symmetric Positive Definite (SPD) manifold, generalising the Euclidean collaborative representation to the SPD manifold is not an easy task. Then, we devise two strategies to cope with this issue. One attempts to embed the SPD manifold-valued data representations into an associated tangent space via the matrix logarithm map. Another is to embed them into a Reproducing Kernel Hilbert Space (RKHS) by utilizing the Riemannian kernel function. After these two treatments, CRC is applicable to the SPD manifold-valued features. The evaluations on four banchmarking datasets justify its effectiveness.