Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models
Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models
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DOI：
10.1214/18-aos1774
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发表时间：
2017-04
期刊：
影响因子：
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通讯作者：
Xin Bing;M. Wegkamp
中科院分区：
文献类型：
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作者：
Xin Bing;M. Wegkamp
我们考虑回归系数矩阵为低阶、未知的多元响应回归问题。在这种情况下，我们分析了一种新的选择最优降阶的准则。该准则与Bunea、She和Wegkamp[7]中提出的准则显著不同，因为它不需要估计噪声的未知方差，也不依赖于微妙的调谐参数选择。我们开发了一种迭代的、完全由数据驱动的程序，以适应最优的信噪比。这个程序在几个步骤中以压倒性的概率找到真正的排名。在每一步，我们的估计都会增加，但同时不会超过真实的排名。我们的有限样本结果适用于任何样本量和任何维度，即使在响应和协变量的数量增长远远快于观察数量的情况下也是如此。我们进行了广泛的模拟研究，证实了我们的理论发现。与文献[7]相比，新方法在低维和高维情况下都有更好的性能和更稳定的性能。
We consider the multivariate response regression problem with a regression coefficient matrix of low, unknown rank. In this setting, we analyze a new criterion for selecting the optimal reduced rank. This criterion differs notably from the one proposed in Bunea, She and Wegkamp [7] in that it does not require estimation of the unknown variance of the noise, nor depends on a delicate choice of a tuning parameter. We develop an iterative, fully data-driven procedure, that adapts to the optimal signal to noise ratio. This procedure finds the true rank in a few steps with overwhelming probability. At each step, our estimate increases, while at the same time it does not exceed the true rank. Our finite sample results hold for any sample size and any dimension, even when the number of responses and of covariates grow much faster than the number of observations. We perform an extensive simulation study that confirms our theoretical findings. The new method performs better and more stable than that in [7] in both low- and high-dimensional settings.