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Establishing correspondences between two feature sets is a fundamental issue in computer vision, pattern recognition, and machine learning. This problem can be well formulated as graph matching in which nodes represent feature points while edges describe pairwise relations between feature points. Recently, several researches have tried to embed higher-order relations of feature points by hyper-graph matching formulations. In this paper, we generalize the previous hyper-graph matching formulations to cover relations of features in arbitrary orders, and propose a novel state-of-the-art algorithm by reinterpreting the random walk concept on the hyper-graph in a probabilistic manner. Adopting personalized jumps with a reweighting scheme, the algorithm effectively reflects the one-to-one matching constraints during the random walk process. Comparative experiments on synthetic data and real images show that the proposed method clearly outperforms existing algorithms especially in the presence of noise and outliers.