Theoretical characterization of search-rank dependence

Characterize theoretically how the factor search rank affects optimization and approximation performance in the rectangular, multi-rate factorization used by the Green's Observation Operator, including the dependence of the attained objective on the search rank.

Background

The Green's Observation Operator represents each temporal kernel mode using a factorization of the coefficient matrix, with the search rank controlling the number of factor columns. The associated factorized objective is generally nonconvex, and the paper explains that existing guarantees for positive-semidefinite Burer–Monteiro factorizations, restricted-isometry matrix sensing, and related low-rank optimization settings do not directly apply to the rectangular, multi-rate structure considered here.

Empirically, increasing the search rank substantially improves the coefficient-space objective up to an apparent elbow: rank one performs much worse than the convex coefficient-space optimum, whereas ranks eight and above approach that optimum. A general theoretical explanation of this observed dependence remains unresolved.