Probabilistic Interpretation Toward Manifold Convex Hull Collaborative Representation

Convex Hull Representation and Learning (CHRL) has drawn great attention for set oriented data analysis. The traditional methods focus on representing set samples in a linear/affine feature space, which can be regarded as a convex geometric region spanned by its feature points. However, the non-Euclidean structure originating from these data is largely ignored by previous CHRLs. To deal with this problem, we propose a new Manifold Convex Hull Collaborative Representation (MCHCR) framework from geometry-aware perspective. Notably, we establish convex hull for Grassmann manifold points modeled by image sets, and then fully exploit the relationships between different sets by collaborative representation mechanism with a closed form solution. Our method definitely inherits the rich representability of Grassmann manifold and high flexibility of convex hull model. Moreover, we interpret the intrinsic classification mechanism of MCHCR under a solid theoretical analysis from probability perspective. Finally, the effectiveness of the proposed learning framework is effectively demonstrated on various benchmark datasets.