[Elroch]

Moobb, having rewatched the Q&A, my understanding is this. The independence that is important is that the input points are independently selected. Intuitively, they are a representative sample, rather than one which gives disproportionate importance to some region of the input space.

With regard to the features, these are a generalisation of co-ordinates which are used to describe the input data points (eg the value of a moving average is a feature which can be thought of as a co-ordinate, even though it is defined in terms of many co-ordinates). The independence that is preserved after a transformation is the independence between the data points, not the features: the set of points remains a representative sample of the (transformed) space of possible inputs.