Mode estimation on matrix manifolds: Convergence and robustness
Mode estimation on matrix manifolds: Convergence and robustness
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发表时间：
2022
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通讯作者：
Hiroaki Sasaki;J. Hirayama;T. Kanamori
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作者：
Hiroaki Sasaki;J. Hirayama;T. Kanamori
矩阵流形上的数据在广泛的研究fi领域中无处不在。关键问题是对数据背后的概率密度函数的模式(即最大值)的估计。例如，局部模式(即，局部极大值)可用于聚类，而全局模式(即，全局极大值)是Fréechet均值的稳健替代。此前，为了估计模式，已经提出了一种基于黎曼梯度估计器的迭代方法，经验表明该方法在聚类中具有优越的性能(Ashizawa等人，2017)。然而，迭代法是否能够捕获基于梯度估计器的模式还没有得到理论上的研究。本文提出了一种基于欧氏度量的矩阵流形上模式估计的简单迭代方法。一个关键的贡献是对所提出的迭代方法进行了理论分析，并建立了单调上升和收敛的充分条件。另外，对于前面的方法，我们证明了模的单调上升性。因此，我们的工作也可以被视为弥补了以前方法中理论分析的不足。此外，还从理论上研究了迭代方法的稳健性。最后，实验证明，该方法在矩阵流形上的聚类和稳健模式估计中具有良好的效果。
Data on matrix manifolds are ubiquitous on a wide range of research fields. The key issue is estimation of the modes (i.e., maxima) of the probability density function underlying the data. For instance, local modes (i.e., local maxima) can be used for clustering, while the global mode (i.e., the global maximum) is a robust alternative to the Fr´echet mean. Previously, to estimate the modes, an iterative method has been proposed based on a Riemannian gradient estimator and empirically showed the superior performance in clustering (Ashizawa et al., 2017). However, it has not been theoretically investigated if the iterative method is able to capture the modes based on the gradient estimator. In this paper, we propose simple iterative methods for mode estimation on matrix manifolds based on the Euclidean metric. A key contribution is to perform theoretical analysis and establish sufficient conditions for the monotonic ascending and convergence of the proposed iterative methods. In addition, for the previous method, we prove the monotonic ascending property towards a mode. Thus, our work can be also regarded as compensating for the lack of theoretical analysis in the previous method. Furthermore, the robustness of the iterative methods is theoretically investigated in terms of the breakdown point. Finally, the proposed methods are experimentally demonstrated to work well in clustering and robust mode estimation on matrix manifolds.