Revisiting the Variable Projection Method for Separable Nonlinear Least Squares Problems

Variable Projection (VarPro) is a framework to solve optimization problems efficiently by optimally eliminating a subset of the unknowns. It is in particular adapted for Separable Nonlinear Least Squares (SNLS) problems, a class of optimization problems including low-rank matrix factorization with missing data and affine bundle adjustment as instances. VarPro-based methods have received much attention over the last decade due to the experimentally observed large convergence basin for certain problem classes, where they have a clear advantage over standard methods based on Joint optimization over all unknowns. Yet no clear answers have been found in the literature as to why VarPro outperforms others and why Joint optimization, which has been successful in solving many computer vision tasks, fails on this type of problems. Also, the fact that VarPro has been mainly tested on small to medium-sized datasets has raised questions about its scalability. This paper intends to address these unsolved puzzles.