Permutation estimation and minimax rates of identifiability

The problem of matching two sets of features appears in various tasks of computer vision and can be often formalized as a problem of permutation estimation. We address this problem from a statistical point of view and provide a theoretical analysis of the accuracy of several natural estimators. To this end, the notion of the minimax matching threshold is introduced and its expression is obtained as a function of the sample size, noise level and dimensionality. We consider the cases of homoscedastic and heteroscedastic noise and carry out, in each case, upper bounds on the matching threshold of several estimators. This upper bounds are shown to be unimprovable in the homoscedastic setting. We also discuss the computational aspects of the estimators and provide some empirical evidence of their consistency on synthetic data-sets.