An Affine Invariant $k$-Nearest Neighbor Regression Estimate
  We design a data-dependent metric in $\mathbb R^d$ and use it to define the $k$-nearest neighbors of a given point. Our metric is invariant under all affine transformations. We show that, with this metric, the standard $k$-nearest neighbor regression estimate is asymptotically consistent under the usual conditions on $k$, and minimal requirements on the input data. 
G\'erard Biau (LPMA, LSTA, DMA, INRIA Paris - Rocquencourt), Luc   Devroye (SOCS), Vida Dujmovic (SCS), Adam Krzyzak (CSE)
math.ST stat.TH