Let's assume we have an "n" dimensional space defined within a hyper-cube bounds and can project points in this space to a lower "m" dimensional one by multiplying it with a matrix of n*m.
Within the low dimensional space I want to apply a search algorithm after which I get a recommendation in this low dimensional space and want to map this recommendation back to the original n dimensional space.
Needless to say, the solution for mapping this low dimensional point to the high dimensional one is not unique. Nevertheless, for this I apply an optimization to find an input of high dimensional space for which the projection will be what I was recommended with in the low dimensional one.
However, I am not sure how to define the bounds within the low dimensional search space so that this task is done in a sensible way.
For example, as shown in the above figure a 2 dimensional plane can be projected on a 1 dimensional line defined by a vector.
In addition, this bound could possibly be defined in two ways which is described by the below figure in contrast to the above one about which I appreciate your recommendations and thoughts.
For example, does it make sense to project the corners and take the most distant ones as bounds? If so, how do I do it in a high dimensional setting!
