Non-commutative distance formula I am trying to find a distance formula or a method that can give the non-commutative distance between two points in a feature space.
Suppose there are two movies represented in an R^n feature space. Now I want that when I try to find the distance/similarity between these movies using the feature vectors, I get different values with respect to which movie is the reference point i.e.,
Dist(Mov1, Mov2) != Dist(Mov2, Mov1)

I know this is slightly vague, but I am trying to understand my thoughts and would like your help. I found that there is a field called non-commutative geometry, but that seems a little too much and the last resort to dive into. Kindly direct my thoughts.
Take for example the Euclidean distance L2, defined by:
$$L_2(x,y) = \left(\sum_{i=1}^{d} (x_i-y_i)^2\right)^{1/2}$$
where $d$ is the vector dimension. You can easily add a term $\alpha \in (0,1)$ and put more weight on the first term, for example:
$$L_2(x,y) = \left(\sum_{i=1}^{d} (\alpha x_i-(1-\alpha)y_i)^2\right)^{1/2}$$
That will certainly be non-cummutative and basically it can be interpreted as putting more weight on the 'reference' vector.
Now, if you ask me, what would be a good use of it, I hardly understand why you would use a non-commutative distance or non-symmetric distance. First of all, it will not be a distance in mathematical sense, since symmetry is required by definition, secondly, many algorithms works with symmetric distances and can't be adapted easily.