Representing image sets and videos with Grassmann manifold has become popular due to its powerful capability to extract discriminative information in machine learning research. However, existing techniques operations on Grassmann manifold are usually suffering from the problem of computational expensive, thus the application range of Grassmann manifold is limited. In this paper, we propose the Fréchet mean-based Grassmann discriminant analysis (FMGDA) algorithm to implement the videos (or image sets) data dimensionality reduction and clustering task. The data dimensionality reduction algorithm proposed by us can not only be used to reduce Grassmann data from high-dimensional data to a relative low-dimensional data, but also to maximize between-class distance and minimize within-class distance simultaneously. Fréchet mean is used to characterize the clustering center of Grassmann manifold space. We further show that the learning problem can be expressed as a trace ratio problem which can be efficiently solved. We designed a detailed experimental scheme to test the performance of our proposed algorithm, and the tests were assessed on several benchmark data sets. The experimental results indicate that our approach leads to a significant improvement over state-of-the-art methods.

中文翻译：

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基于 Fréchet 均值的 Grassmann 判别分析

用 Grassmann 流形表示图像集和视频已经变得流行，因为它具有在机器学习研究中提取判别信息的强大能力。然而，现有的对 Grassmann 流形的操作通常存在计算量大的问题，因此 Grassmann 流形的应用范围受到限制。在本文中，我们提出了基于 Fréchet 均值的 Grassmann 判别分析 (FMGDA) 算法来实现视频（或图像集）数据降维和聚类任务。我们提出的数据降维算法不仅可以将 Grassmann 数据从高维数据降维到相对低维数据，还可以同时最大化类间距离和最小化类内距离。Fréchet 均值用于刻画 Grassmann 流形空间的聚类中心。我们进一步表明，学习问题可以表示为可以有效解决的迹比问题。我们设计了一个详细的实验方案来测试我们提出的算法的性能，并在几个基准数据集上对测试进行了评估。实验结果表明，我们的方法比最先进的方法有了显着的改进。