近年来, 深度学习以其强大的非线性计算能力在目标检测和识别任务中取得了巨大的突破。现有的深度学习网络几乎都是以数据的欧氏结构为前提, 而在计算机视觉中许多数据都具有严格的流形结构, 如图像集可表示为Grassmann流形。基于数据的流形几何结构来设计深度学习网络, 将微分几何理论与深度学习理论相结合, 提出一种基于Grassmann流形的深度图像集识别网络。同时在模型训练过程中, 使用基于矩阵链式法则的反向传播算法来更新模型, 并将权值的优化过程转换为Grassmann流形上的黎曼优化问题。实验结果表明: 该方法不仅在结果上识别准确率得到了提高, 同时在训练和测试速度上也有一个数量级的提升。

In recent years, deep learning techniques have achieved great breakthrough for its powerful non-linear computations in the tasks of target recognition and detection. Existing deep networks were almost designed based on the precondition that the visual data reside on the Euclidean space. However, many data in computer vision have rigorous geometry of manifolds, i.e., image sets can be represented as Grassmann manifolds. The deep network was devised based on the non-Euclidean structure of the manifold-valued data, which combined the differential geometry and deep learning methods theoretically. Furthermore, a deep network for image-set recognition based on the Grassmann manifold was proposed. In the training process, the model was updated by the use of the backpropagation algorithm derived from the matrix chain rule. Learning of the weights can be transformed as the Riemannian optimization problem on the Grassmannian. The experimental results show that this method not only improves the accuracy of recognition, but also accelerates the training and test process in one magnitude.