Concentration of measure bounds for matrix-variate data with missing values

Abstract

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.