Theoretical analysis of cross-validation for estimating the risk of the k-Nearest Neighbor classifier

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 \cite{CelisseMaryHuard11}. 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

Total Synthesis of (−)-Ecklonialactone B

The total synthesis of (−)-ecklonialactone B as well as the 9,10-dihydro derivative by two different strategies is reported. The catalytic asymmetric Claisen rearrangement of Gosteli-type allyl vinyl ethers delivered elaborated α-keto ester building blocks. Ring-closing metatheses, including a notable diastereotopos-differentiating variant, a <i>B</i>-alkyl Suzuki–Miyaura cross-coupling reaction and a regio- and diastereoselective last-step epoxidation are key contributors