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We consider the following data perturbation model, where the covariates incurmultiplicative errors. For two $n \times m$ random matrices $U, X$, we denoteby $U \circ X$ the Hadamard or Schur product, which is defined as $(U \circX)_{ij} = (U_{ij}) \cdot (X_{ij})$. In this paper, we study the subgaussianmatrix variate model, where we observe the matrix variate data $X$ through arandom 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 ${\mathbbE} U_{ij} \in [0, 1]$ and all entries are mutually independent. Subsampling inrows, or columns, or random sampling of entries of $X$ are special cases ofthis model. Under the assumption of independence between $U$ and $X$, weintroduce componentwise unbiased estimators for estimating covariance $A$ and$B$, and prove the concentration of measure bounds in the sense of guaranteeingthe restricted eigenvalue($\textsf{RE}$) conditions to hold on the unbiasedestimator for $B$, when columns of data matrix $X$ are sampled with differentrates. We further develop multiple regression methods for estimating theinverse of $B$ and show statistical rate of convergence. Our results provideinsight for sparse recovery for relationships among entities (samples,locations, items) when features (variables, time points, user ratings) arepresent in the observed data matrix ${\mathcal X}$ with heterogeneous rates.Our proof techniques can certainly be extended to other scenarios. We providesimulation evidence illuminating the theoretical predictions.

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