A　primary　research　direction　for　recommender　systems　is　matrix　completion,　which　attempts　to　recover　the　missing　values　in　a　user-item　rating　matrix.　There　are　numerous　approaches　for　rating　tasks,　which　are　mainly　classified　into　latent　factor　models　and　neighborhood-based　models.　Most　neighborhood-based　models　seek　similar　neighbors　by　computing　similarities　in　the　original　data　space　for　final　predictions.　In　this　paper,　we　propose　a　new　neighborhood-based　interpolation　model　with　a　kernelized　matrix　completion　framework,　with　the　impact　weights　provided　by　neighbors　computed　in　a　new　Hilbert　space　containing　more　features.　In　our　model,　the　kernel　function　is　combined　with　a　similarity　measurement　to　achieve　better　approximation　for　unknown　ratings.　Furthermore,　we　extend　our　model　with　a　non-linear　multi-kernel　framework　which　learns　weights　automatically　to　improve　the　model.　Finally,　we　conduct　extensive　experiments　on　several　real-world　datasets.　The　outcomes　show　that　the　proposed　methods　work　effectively　and　improve　the　performance　of　the　rating　prediction　task　compared　to　both　the　traditional　and　state-of-the-art　approaches.