Finding Feature Representation Such That Two Samples Are Similar in Feature Space Consider one specific useful function of our human brain: abstraction of object. Take the example of two pictures: if we are told the pictures are similar, we actually make conclusion about the aspects in which they are close to each other.

I'm considering whether machine can have the ability described. More accurately, is it possible to find and select a set of feature representations of two samples (e.g. image, sound) such that under those representations, the samples are similar with respect to a metric, say weighted euclidean norm?
I think you're looking for metric learning or Manifold learning. At a very high level, the idea behind both of these approaches is to learn the space (or transform) over which a set of (labeled) examples are close to one another.