Robust Grassmann manifold convex hull collaborative representation learning and its kernel extension for image set analysis

Effectively leveraging multi-view information is crucial for in-depth analysis of complex problems. Currently, the approach of analyzing sets of images has garnered significant attention, mainly because it allows for the comprehensive representation of a subject through the integration of multiple images. Previous image set classification methods mainly focus on deriving collaborative representation models on Euclidean space to enhance the ability of feature learning. However, these methods often neglect the underlying manifold geometry structure of the image set. In this paper, we propose a novel robust grassmann manifold convex hull collaborative representation (RGMCHCR) framework with L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm regularization from geometry-aware perspective. Our model achieves the goal of inheriting the highly expressive representation capability of the Grassmann manifold, while also maintaining the flexible nature of the convex hull model. Notably, the collaborative representation mechanism places a strong emphasis on exploring connections between diverse convex hulls within the Grassmann manifold. Besides, we regularize the collaborative representation coefficients by using the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm, showcasing superior noise robustness and meeting the requirements for data reconstruction. Furthermore, we derive the Kernelized RGMCHCR to better model complex nonlinear problems. Through extensive experiments and comprehensive comparisons, our method's effectiveness in image set classification surpasses that of other existing approaches.