Can I find a mapping that minimizes the maximum distance ratio of certain vectors? Let's say we have several vector points. My goal is to distinguish the vectors, so I want to make them far from each other.  Some of them are already far from each other, but some of them can be positioned very closely.  

I want to get a certain mapping function that can separate such points that are close to each other, while still preserving the points that are already far away from each other. 

I do not care what is the form of the mapping. Since the mapping will be employed as pre-processing, it does not have to be differentiable or even continuous.  

I think this problem is somewhat similar to 'minimizing the maximum distance ratio between the points'. Maybe this problem can be understood as stretching the crushed graph to a sphere-like isotropic graph. 

I googled it for an hour, but it seems that the people are usually interested in selecting the points that have such nice characteristics from a bunch of data, rather than mapping an existing vector points to a better one.

So, in conclusion, I could not find anything useful.   

Maybe you can think 'the neural network will naturally learn it while solving classification problem'. But it failed. Because it is already struggling with too many burdens. So, this is why I want to help my network with pre-processing.
An interesting question. I would start by finding n nearest neighbors of each data point, then calculate their center of mass c and the point's distance d to its nth nearest neighbor. The smaller the d is the larger the density is around a given point. You could then iteratively step every point away from their c in an inverse proportion to the distance d with a suitable step size. This would spread out the clusters.

But this won't help you transform any new points in the dataset, maybe you can learn this arbitrary mapping R^n -&gt; R^n by using an other neural network and apply it to new samples?

This is the first ad-hoc idea which came to my mind. It would be interesting to see a 2D animation of this.

A more rigorous approach might be a variational autoencoder, you can embed the data in a lower dimensional space with approximately normal distribution. But it doesn't guarantee that clusters would be as spread out as you'd like. An alternative loss function would help with that, for example every point's distance to their original nth closest neighbor should be as close to one as possible.