# On Inferences from Completed Data

## TL;DR

This paper studies how errors from matrix completion impact statistical inference, providing error bounds and demonstrating that perfect matrix recovery isn't always necessary for accurate inference.

## Contribution

It introduces recovery error bounds for statistical inference based on matrix completion and analyzes the effects of approximate recovery in practical scenarios.

## Key findings

- Error bounds depend on matrix recovery error.
- Exact matrix recovery isn't always needed for accurate inference.
- Numerical experiments confirm theoretical insights.

## Abstract

Matrix completion has become an extremely important technique as data scientists are routinely faced with large, incomplete datasets on which they wish to perform statistical inferences. We investigate how error introduced via matrix completion affects statistical inference. Furthermore, we prove recovery error bounds which depend upon the matrix recovery error for several common statistical inferences. We consider matrix recovery via nuclear norm minimization and a variant, $\ell_1$-regularized nuclear norm minimization for data with a structured sampling pattern. Finally, we run a series of numerical experiments on synthetic data and real patient surveys from MyLymeData, which illustrate the relationship between inference recovery error and matrix recovery error. These results indicate that exact matrix recovery is often not necessary to achieve small inference recovery error.