Self-adaptive subspace representation from a geometric intuition

In this paper, we address the unsupervised subspace learning task from a geometric intuition, which offers a direct understanding of the data representation. First, we consider all subspaces of the Euclidean space as a series of graded Grassmannian manifolds, and represent them by orbits of rotation group action. Then, we reformulate the unsupervised subspace learning by a least square problem with respect to rotation and projection operator. Second, we introduce a low-rank regularization to obtain a low-dimensional and robust subspace representation. Then, the model is translated into a minimization problem on the Grassmannian manifold. By dividing the model into two subproblems of solving the optimal rotation and subspace dimension, we design an alternating iteration strategy, where the locally geodesic structure of the rotation group and the unconstrained quadratic 0-1 programming are utilized. Finally, we apply the proposed method to the image classification problem on five benchmark datasets and compare it with nine state-of-the-art methods. Numerical results show that our proposed method has better feature representation ability and almost achieves the best performance in terms of classification accuracy and robustness.