This　paper　investigates　algorithms　for　matrix　factorization　when　some　or　many　components　are　missing,　a　problem　that　arises　frequently　in　computer　vision　and　pattern　recognition.　We　demonstrate　that　the　Jacobian　used　in　the　damped　Wiberg　(DW)　method　is　exactly　the　same　as　that　of　Kaufman＇s　simplified　variable　projection　(VP)　algorithm.　Our　analysis　provides　a　novel　perspective　on　the　efficiency　of　VP　algorithms　by　improving　the　strong　convexity　of　the　approximate　function.　To　enhance　numerical　stability,　we　set　a　lower　bound　on　the　damping　parameter　instead　of　adding　a　null　space　like　the　DW　algorithm.　Another　challenge　of　low-rank　matrix　decomposition　with　missing　data　is　the　existence　of　many　sharp　local　minima,　which　are　often　distributed　in　narrow　valleys　of　the　landscape　of　objection　functions.　Falling　into　such　minima　tends　to　result　in　poor　reconstruction　results.　To　address　this　issue,　we　design　a　non-monotonic　VP　algorithm,　which　can　facilitate　the　algorithm　to　escape　from　sharp　minima　and　converge　to　flatter　minima.　Numerical　experiments　confirm　the　effectiveness　and　efficiency　of　the　proposed　nonmonotone　VP　algorithm.　©　2023