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info.json
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{
"abstract": "The present work aims at deriving theoretical guaranties on the behavior of some cross-validation procedures applied to the $k$-nearest neighbors ($k$NN) rule in the context of binary classification. Here we focus on the leave-$p$-out cross-validation (L$p$O) used to assess the performance of the $k$NN classifier. Remarkably this L$p$O estimator can be efficiently computed in this context using closed-form formulas derived by Celisse and Mary-Huard (2011). We describe a general strategy to derive moment and exponential concentration inequalities for the L$p$O estimator applied to the $k$NN classifier. Such results are obtained first by exploiting the connection between the L$p$O estimator and U-statistics, and second by making an intensive use of the generalized Efron-Stein inequality applied to the L$1$O estimator. One other important contribution is made by deriving new quantifications of the discrepancy between the L$p$O estimator and the classification error/risk of the $k$NN classifier. The optimality of these bounds is discussed by means of several lower bounds as well as simulation experiments.",
"authors": [
"Alain Celisse",
"Tristan Mary-Huard"
],
"id": "15-498",
"issue": 58,
"pages": [
1,
54
],
"title": "Theoretical Analysis of Cross-Validation for Estimating the Risk of the $k$-Nearest Neighbor Classifier",
"volume": 19,
"year": 2018
}