具有缺失值的矩阵变量数据的测度界的集中性
Concentration of measure bounds for matrix-variate data with missing values
摘要：
我们考虑下面的数据扰动模型，其中协变量产生乘法误差。对于两个$n \times m$随机矩阵$U, X$，我们用$U \circ X$表示Hadamard或Schur乘积，定义为$(U \circ X)_{ij} = (U_{ij}) \cdot (X_{ij})$。本文研究了亚高斯矩阵变量模型，其中我们通过随机掩码$U$观察矩阵变量数据$X$：$$ {\mathcal X} = U \circ X \; \; \; \text{ where} \; \; \;X = B^{1/2} {\mathbb{Z}} A^{1/2}, $$其中${\mathbb{Z}}$是具有独立亚高斯元素的随机矩阵，$U$是具有零或正元素的掩码矩阵，其中${\mathbb E} U_{ij} \in [0, 1]$和所有元素相互独立.行或列中的二次抽样或$X$条目的随机抽样是此模型的特殊情况。在$U$和$X$相互独立的假设下，引入了估计协方差$A$和$B$的分量无偏估计，并证明了当数据矩阵$X$的列以不同的采样率采样时，在保证约束特征值（$\textsf{RE}$）条件的意义下，$B$的无偏估计保持测度界的集中性.我们进一步发展多元回归方法估计逆$B$和统计收敛速度。我们的研究结果提供了洞察稀疏恢复实体（样本，位置，项目）之间的关系时，功能（变量，时间点，用户评级）存在于所观察到的数据矩阵${\mathcal X}$异构率。我们的证明技术当然可以扩展到其他场景。我们提供模拟证据照亮理论预测。
英文摘要：
We consider the following data perturbation model, where the covariates incur multiplicative errors. For two $n \times m$ random matrices $U, X$, we denote by $U \circ X$ the Hadamard or Schur product, which is defined as $(U \circ X)_{ij} = (U_{ij}) \cdot (X_{ij})$. In this paper, we study the subgaussian matrix variate model, where we observe the matrix variate data $X$ through a random mask $U$: $$ {\mathcal X} = U \circ X \; \; \; \text{ where} \; \; \;X = B^{1/2} {\mathbb{Z}} A^{1/2}, $$ where ${\mathbb{Z}}$ is a random matrix with independent subgaussian entries, and $U$ is a mask matrix with either zero or positive entries, where ${\mathbb E} U_{ij} \in [0, 1]$ and all entries are mutually independent. Subsampling in rows, or columns, or random sampling of entries of $X$ are special cases of this model. Under the assumption of independence between $U$ and $X$, we introduce componentwise unbiased estimators for estimating covariance $A$ and $B$, and prove the concentration of measure bounds in the sense of guaranteeing the restricted eigenvalue($\textsf{RE}$) conditions to hold on the unbiased estimator for $B$, when columns of data matrix $X$ are sampled with different rates. We further develop multiple regression methods for estimating the inverse of $B$ and show statistical rate of convergence. Our results provide insight for sparse recovery for relationships among entities (samples, locations, items) when features (variables, time points, user ratings) are present in the observed data matrix ${\mathcal X}$ with heterogeneous rates. Our proof techniques can certainly be extended to other scenarios. We provide simulation evidence illuminating the theoretical predictions.
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