We consider the use of data reduction techniques for the problem of approximate query answering. We focus on applications for which accurate answers to selective queries are required, and for which the data are very high dimensional (having hundreds or perhaps thousands of dimensions). We carefully examine the assumptions underlying many existing reduction techniques. To ensure both speed and accuracy, we show that these methods assume statistical characteristics that very high dimensional datasets do not in general possess. We present a new data reduction method that does not suffer from these limitations, called the RS Kernel. We demonstrate the effectiveness of this method for answering difficult, highly selective queries over high dimensional data using several real datasets.