One　of　the　main　methods　for　dealing　with　low-rank　problems　is　the　matrix　completion,　which　attempts　to　recover　incomplete　values　in　an　incomplete　matrix.　It　has　been　employed　in　a　large　number　of　real-world　applications　like　recommender　systems　and　image　recovery.　When　the　data　is　highly　sparse,　many　practical　problems　of　recommender　systems　and　image　recovery　are　hard　to　solve.　Therefore,　we　use　the　fully　connected　neural　networks　for　the　matrix　completion　to　avoid　problems　caused　by　sparse　matrices.　In　this　paper,　we　utilize　the　local　density　loss　function　to　measure　the　difference　of　the　matrix　completion　results,　where　trainable　parameters　are　updated　via　calculating　the　derivatives　according　to　the　influence　function.　The　local　density　loss　function　effectively　measures　the　deviation　between　the　predicted　value　and　the　real　value,　and　the　convergence　of　the　model　is　guaranteed.　So　as　to　validate　the　effectiveness　of　the　proposed　method,　we　conduct　substantial　experiments　in　terms　of　image　recovery　and　recommender　systems.　In　addition,　we　employ　three　metrics,　including　root　mean　square　error　and　peak　signal-to-noise　ratio　and　structure　similarity,　to　measure　the　recovery　accuracy　of　missing　matrix.　Experimental　results　demonstrate　that　this　framework　is　superior　to　other　state-of-the-art　methods　in　running　time　and　learning　performance.　©　2022　SPIE.