> there are four projections of the 50-dimensional vector space that would project out just as well from a random matrix because, well, projecting 20 generic po

> there are four projections of the 50-dimensional vector space that would project out just as well from a random matrix because, well, projecting 20 generic points in a 50-dimensional space to 2-dimensions you can make the points fall exactly where you want in 2 dimensions. Well, no. Linear projections follow a bunch of rules that enforce conservation of various linear structures. You can't manipulate things arbitrarily. For example, if three points are colinear in the original space, they will be colinear in every projection. Perhaps more relevant to the GloVe examples, if a-b = c-d in the original space then the same equality holds in every projection. Since the projections are Lipschitz continuous, we can also say that is a-b is close to c-d in the original space then they are also close in every projection. Anyway, the point that these figures illustrate is easily confirmed by downloading the embeddings yourself, so insinuating that the authors are getting away with something that they should be held to account for is silly.