We propose an efficient and robust iterative solution to the multi-object matching problem. We first clarify serious limitations of current methods as well as the inappropriateness of the standard iteratively reweighted least squares procedure. In view of these limitations, we suggest a novel and more reliable iterative reweighting strategy that incorporates information from higher-order neighborhoods by exploiting the graph connection Laplacian. We provide partial theoretical guarantees and demonstrate the superior performance of our procedure over state-of-the-art methods using both synthetic and real datasets.