【摘要】 实际应用中经常会遇到误差数据的情形,例如生物信息学,神经影像和遥感研究等.现有方法主要考虑线性或广义线性回归的变量误差模型,而较少关注多响应回归的情形,并且如何评价协变量带有扰动时的估计性能,亦即信息理论极限,这仍然是一个有待解决的问题.本文考虑了在高维多响应回归模型中估计一类低秩矩阵的信息理论极限.应用信息理论和关于集中不等式的统计技巧,本文以平方Frobenius损失函数的形式给出了极小极大下界,这一下界达到了以往文献在干净协变量假设下的收敛率.这一结果进一步表明即使在更具现实意义下的变量误差情形中,仍然不需要更多的样本以获得收敛率最优的估计.更多还原

【Abstract】 noisy data is always encountered in real applications, such as bioinformatics, neuroimage and remote sensing. Existing methods mainly consider linear or generalized linear errors-in-variables regression, while relatively little attention is paid for the multivariate response case, and how to evaluate the estimation performance, i.e., the information-theoretic limitation, under perturbed covariates is still an open question. In this paper, the information-theoretic limitation of estimating a low-rank matrix in the multi-response errors-in-variables regression model is considered. By application of information theory and statistical techniques on concentration inequalities, the minimax lower bound is provided in terms of the squared Frobenius loss, which recaptures the rate provided under the clean covariate assumption in previous literatures. Hence our result further indicates that though under the more realistic errors-in-variables situation, no more samples are required so as to achieve a rate-optimal estimation.更多还原