Representing the data on Grassmann manifolds is popular in quite a few image and video recognition tasks. In order to enable deep learning on Grassmann manifolds, this paper proposes a deep network architecture which generalizes the Euclidean network paradigm to Grassmann manifolds. In particular, we design full rank mapping layers to transform input Grassmannian data into more desirable ones, exploit orthogonal re-normalization layers to normalize the resulting matrices, study projection pooling layers to reduce the model complexity in the Grassmannian context, and devise projection mapping layers to turn the resulting Grassmannian data into Euclidean forms for regular output layers. To train the deep network, we exploit a stochastic gradient descent setting on manifolds where the connection weights reside on, and study a matrix generalization of backpropagation to update the structured data. We experimentally evaluate the proposed network for three computer vision tasks, and show that it has clear advantages over existing Grassmann learning methods, and achieves results comparable with state-of-the-art approaches.

本文提出了一种将欧几里得网络范例泛化到Grassmann 流形的深度网络体系结构，利用完全秩映射层将输入数据转换为更理想的数据，利用再正则化层对结果矩阵进行规范化，利用投影映射层使结构数据进行更新，并在三个视觉识别任务中的评估显示出 Grassmann 网络具有明显优势。

在 Grassmann 流形上构建深度神经网络