I have a set of objects. I have calculated two distance matrices: $X$ defining distance between each objects pair using metric $f1$, and $Y$ -- using metric $f2$. Now, I would like to understand if two objects are similar according to metric $f1$, then they are also similar according to metric $f2$. How can I do it?
For instance, $f1$ could say whether two objects have similar color, and $f2$ --- whether two objects have similar size. But metrics can be anything. For instance, we could talk about articles, $f1$ could be Jaccard distance measuring how many tags both articles share, and $f2$ could be euclidian distance measuring distance between word vectors of two articles. Now I would like to understand if two objects of blueish objects tend to be big, or whether articles tagged with "racism" have similar content.
Am I asking about correlation? How can I calculate it between $X$ and $Y$?