Interesting. IIUC, what you're implying is that defining a metric defines the topology and they're equivalent.
Isn't p_ij in t-SNE also derived from the distances themselves, where p_ij ~ student_t(d_ij, degrees_of_freedom) (I forget how the d.o.f. is actually computed in t-SNE.)
Which leads me to one way this distance based approach might be limited: It models similarities using distances, which are symmetric. If similarities aren't symmetric, then this visualisation could hide some information. For example: The specific entity "BMW car" is more similar to the more general entity "car" than the entity "car" is to "BMW car." It seems this asymmetry could capture things (such as the generality of concepts), not reflected in metric spaces (on first thought).
Isn't p_ij in t-SNE also derived from the distances themselves, where p_ij ~ student_t(d_ij, degrees_of_freedom) (I forget how the d.o.f. is actually computed in t-SNE.)
Which leads me to one way this distance based approach might be limited: It models similarities using distances, which are symmetric. If similarities aren't symmetric, then this visualisation could hide some information. For example: The specific entity "BMW car" is more similar to the more general entity "car" than the entity "car" is to "BMW car." It seems this asymmetry could capture things (such as the generality of concepts), not reflected in metric spaces (on first thought).