Select the correct rank among multiple exact-completion outputs

Determine a rigorous procedure for selecting the correct recovered matrix when alternating-projection or related matrix-completion algorithms are run for multiple candidate ranks and produce distinct outputs that agree on the observed entries.

Background

The paper discusses the practical situation in which the exact rank of the unknown matrix is unavailable, but lower and upper estimates for the rank are known. A natural proposal is to run the recovery algorithm for every candidate rank in the range. Although this increases the running time by only a moderate multiplicative factor, the resulting outputs may not be uniquely identifiable from the observed entries.

The authors specifically identify the unresolved issue of how to choose among different outputs that agree on the observation set when exact recovery is required. This is a methodological gap concerning model selection and uniqueness, rather than a limitation of a particular numerical implementation.