So far, I have read some highly cited metric learning papers. The general idea of such papers is to learn a mapping such that mapped data points with same label lie close to each other and far from samples of other classes. To evaluate such techniques they report the accuracy of the KNN classifier on the generated embedding. So my question is if we have a labelled dataset and we are interested in increasing the accuracy of classification task, why do not we learn a classifier on the original datapoints. I mean instead of finding a new embedding which suites KNN classifier, we can learn a classifier that fits the (not embedded) datapoints. Based on what I have read so far the classification accuracy of such classifiers is much better than metric learning approaches. Is there a study that shows metric learning+KNN performs better than fitting a (good) classifier at least on some datasets?