Assume I have four-vectors (v1,v2,v3,v4), and I want to create a new vector (vec_new) that is not close to any of those four-vectors. I was thinking about interpolation and extrapolation. Do you think they are suitable? Are they also apply for vector and generate a vector of let's say 300 dimensions? Another possible option would be the transformation matrix. But I am not sure if it fit my concern. I think averaging and concatenation are not the good ones as I might be close to some of those four-vectors. I am considering the cosine similarity as the measurement to find the closeness.
I got the vectors from the word2vec model. Based on some criteria, I split them into two categories. The splitting process is not important at all. The only important point is that there are some categories, and the vectors of that category are useful for that category. My concern is to add a new category that the vectors of this category have two conditions. First, the new category vectors are based on those two categories vectors. Second, new category vectors are as far as possible from those two categories vectors. There is no restriction for the procedure, but it should be valid and generalizable.
Any other ideas?
For a general case, if the given vectors span a subpace with dimension less than the dimension of the space, a possible way to create a new vector "not close" to the others is to take $v_5$ orthogonal to the span of the given vectors.