STATISTICS ON THE STIEFEL MANIFOLD: THEORY AND APPLICATIONS
STATISTICS ON THE STIEFEL MANIFOLD: THEORY AND APPLICATIONS
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DOI：
10.1214/18-aos1692
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发表时间：
2019-02-01
影响因子：
4.5
通讯作者：
Vemuri, Baba C.
中科院分区：
文献类型：
--
作者：
Chakraborty, Rudrasis;Vemuri, Baba C.
紧凑型Stiefel流形经常出现在许多工程领域，包括信号和图像处理、机器学习、数值优化等。Stiefel流形是黎曼齐性空间，但不是对称空间。在以前的工作中，研究人员定义了对称空间上的概率分布，并对这些空间中的数据进行了统计分析。在本文中，我们提出了原来的工作涉及定义的高斯分布在齐次空间，并表明，最大似然估计的位置参数的高斯分布在齐次空间产生的Frechet平均（FM）的样本从这个分布。此外，我们提出了一个算法，从高斯分布的Stiefel流形上采样，并递归计算这些样本的FM。我们还证明了这种递归FM估计的弱相合性。几个合成和真实的数据实验，然后提出，证明了上级的计算性能，这种估计的梯度下降为基础的非递归计数器的一部分，以及随机梯度下降为基础的方法在文献中流行。
A Stiefel manifold of the compact type is often encountered in many fields of engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data residing in these spaces. In this paper, we present original work involving definition of Gaussian distributions on a homogeneous space and show that the maximum-likelihood estimate of the location parameter of a Gaussian distribution on the homogeneous space yields the Frechet mean (FM) of the samples drawn from this distribution. Further, we present an algorithm to sample from the Gaussian distribution on the Stiefel manifold and recursively compute the FM of these samples. We also prove the weak consistency of this recursive FM estimator. Several synthetic and real data experiments are then presented, demonstrating the superior computational performance of this estimator over the gradient descent based nonrecursive counter part as well as the stochastic gradient descent based method prevalent in literature.