Test Point=c(1,0) Test Point=c(1,0)

The multi-class metric problem in nearest neighbour dis-21 nearest neighbors than we achieved. Friedman (1994) proposes a number of techniques for exible metric nearest neighbor classiication. These techniques use a recursive partitioning style strategy to adaptively shrink and shape rectangular neighborhoods around the test point. Friedman also uses derived variables in the process, including discriminant variates. With the latter variables, his procedures have some similarity to the discriminant adaptive nearest neighbor approach. Other recent work that is somewhat related to this is that of Lowe (1993). He estimates the covariance matrix in a variable kernel classiier using a neural network approach. There are a number of ways in which this work might be generalized. In some discrimination problems, it is natural to use specialized distance measures that capture invariances in the feature space., use a transformation-invariant metric to measure distance between digitized images of handwritten numerals in a nearest neighbor rule. The invariances include local transformations of images such as rotation, shear and stroke-thickness. An invariant distance measure might be used in a linear discriminant analysis and hence in the DANN procedure. Another interesting possibility would be to apply the techniques of this paper to regression problems. In this case the response variable is quantitative rather than a class label. Natural analogues of the local between and within matrices exist, and can be used to shape the neighborhoods for near-neighbor and local polynomial regression techniques. Likewise, the dimension reduction ideas of section 3 can also be applied. There is a strong connection between the latter and the Sliced Inverse Regression technique of Duan & Li (1991) for subspace identiication. We are currently exploring these directions. Acknowledgements We thank Jerome Friedman for sharing his recent work, which stimulated us to embark on this project, and for many enjoyable conversations. 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 # of noise vars