噪声和高维尺度下（近）低秩矩阵的估计
Estimation of (near) low-rank matrices with noise and high-dimensional scaling
摘要：
高维推断指的是统计估计问题，其中数据的环境维度可能与样本量相当或甚至可能大于样本量。我们研究了一个高维推理的例子，其目标是估计一个矩阵$Θ^* \in \real^{k \times p}$的基础上$N$噪声观测，和未知矩阵$Θ^*$被假定为要么是完全低秩，或"近“低秩，这意味着它可以很好地近似的矩阵与低秩。我们考虑了一个基于矩阵迹或核范数正则化的$M$ -估计器，并分析了它在高维尺度下的性能。我们提供了非渐近界的Frobenius范数误差，持有一般类的噪声观测模型，然后说明其后果的一些特定的矩阵模型，包括低秩多变量或多任务回归，向量自回归过程中的系统辨识，和恢复低秩矩阵随机投影。仿真结果表明，与我们的理论预测的误差的高维标度非常吻合。
英文摘要：
High-dimensional inference refers to problems of statistical estimation in which the ambient dimension of the data may be comparable to or possibly even larger than the sample size. We study an instance of high-dimensional inference in which the goal is to estimate a matrix $Θ^* \in \real^{k \times p}$ on the basis of $N$ noisy observations, and the unknown matrix $Θ^*$ is assumed to be either exactly low rank, or ``near'' low-rank, meaning that it can be well-approximated by a matrix with low rank. We consider an $M$-estimator based on regularization by the trace or nuclear norm over matrices, and analyze its performance under high-dimensional scaling. We provide non-asymptotic bounds on the Frobenius norm error that hold for a general class of noisy observation models, and then illustrate their consequences for a number of specific matrix models, including low-rank multivariate or multi-task regression, system identification in vector autoregressive processes, and recovery of low-rank matrices from random projections. Simulation results show excellent agreement with the high-dimensional scaling of the error predicted by our theory.
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