RATE-OPTIMAL PERTURBATION BOUNDS FOR SINGULAR SUBSPACES WITH APPLICATIONS TO HIGH-DIMENSIONAL STATISTICS
RATE-OPTIMAL PERTURBATION BOUNDS FOR SINGULAR SUBSPACES WITH APPLICATIONS TO HIGH-DIMENSIONAL STATISTICS
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DOI：
10.1214/17-aos1541
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发表时间：
2018-02-01
影响因子：
4.5
通讯作者：
Zhang, Anru
中科院分区：
文献类型：
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作者：
Cai, T. Tony;Zhang, Anru
奇异空间的扰动界，特别是Wedin的sin Theta定理，是高维统计、机器学习和应用数学等许多领域的基本工具。在本文中，我们建立单独的扰动界，测量在频谱和Frobenius sin θ距离，左，右奇异子空间。同时给出了各个扰动界的下界，这些下界表明各个扰动界是速率最优的，新的扰动界适用于更广泛的问题。在本文中，我们详细考虑应用低秩矩阵去噪和奇异空间估计，高维聚类和典型相关分析（CCA）。特别是，单独的匹配上界和下界估计的左，右奇异空间。据我们所知，这是第一个结果，给出了不同的最佳率的左，右奇异空间在相同的扰动。
Perturbation bounds for singular spaces, in particularWedin's sin Theta theorem, are a fundamental tool in many fields including high-dimensional statistics, machine learning and applied mathematics. In this paper, we establish separate perturbation bounds, measured in both spectral and Frobenius sin Theta distances, for the left and right singular subspaces. Lower bounds, which show that the individual perturbation bounds are rate-optimal, are also given.The new perturbation bounds are applicable to a wide range of problems. In this paper, we consider in detail applications to low-rank matrix denoising and singular space estimation, high-dimensional clustering and canonical correlation analysis (CCA). In particular, separate matching upper and lower bounds are obtained for estimating the left and right singular spaces. To the best of our knowledge, this is the first result that gives different optimal rates for the left and right singular spaces under the same perturbation.