Statistical Inferences of Linear Forms for Noisy Matrix Completion
This work addresses the need for reliable statistical inference in noisy matrix completion, which is important for applications like recommendation systems and data imputation, though it is incremental as it builds on existing entry-wise consistent estimators.
The authors tackled the problem of making statistical inferences about linear forms of a matrix from noisy, incomplete observations, developing a universal procedure that yields asymptotically normal estimators and enables confidence intervals and hypothesis testing, with practical validation on simulated and real-world data.
We introduce a flexible framework for making inferences about general linear forms of a large matrix based on noisy observations of a subset of its entries. In particular, under mild regularity conditions, we develop a universal procedure to construct asymptotically normal estimators of its linear forms through double-sample debiasing and low-rank projection whenever an entry-wise consistent estimator of the matrix is available. These estimators allow us to subsequently construct confidence intervals for and test hypotheses about the linear forms. Our proposal was motivated by a careful perturbation analysis of the empirical singular spaces under the noisy matrix completion model which might be of independent interest. The practical merits of our proposed inference procedure are demonstrated on both simulated and real-world data examples.