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What is the word describing a feature space where distance between two elements has a decisive informational value, whatever the pair of elements is?

For example if a model creates embeddings for words, and we take all possible pairs of embeddings and compute their distances, and it is possible to set a fixed threshold distance, where every pair whose distance is inferior to the threshold are synonyms, and every pair with distance superior to the threshold aren't. Is there an adjective that qualify a given feature space that has this property?

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A vector space can be named based on the properties of functions that can be defined in it.

If a vector space is structured enough to contain a metric (e.g. euclidean distance or generic inner product) then we can call that space a Metric Space.

In the case of cosine similarity, we can't talk about metric space since cosine similarity is not a metric (it doesn't respect the triangular inequality), but it is a valid dissimilarity function, so you could refer to that vector space as a Dissimilarity Space.

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    $\begingroup$ I am sorry, but my question is more constrained than that. Given a metric space, what is the name of the subset of metric spaces where the given distance metric has the property such as described in the original question ? The second paragraph is not just a vague description, every word of it counts. $\endgroup$
    – Sorenai de
    Mar 24, 2022 at 8:37
  • $\begingroup$ well, again it really depends on the function you define. What you describe sounds to me like a dissimilarity function. So if you can provide a rigorous definition of it and prove that it satisfies the properties of a dissimilarity function (i.e. the more dissimilar two words are the higher the value returned by the function is) then you can call that space a dissimilarity space. Otherwise the description of such space is too generic to associate it to a specific nomenclature. $\endgroup$ Mar 24, 2022 at 9:35
  • $\begingroup$ I will try to rephrase it. One way to use a feature space equipped with a distance metric is to take a specific element A, and sort every other element by distance to this element. You would obtain for example the distance list [0.1, 2, 5, 7,...]. Then you would do the same thing with another element B and get the list [10, 20, 30, 70,...]. A feature space that would NOT have the property i am talking about would be if the closest element to element A is of the same identity, AND the the closest element to element B is of the same identity. ie, you cant set a decisive distance threshold. $\endgroup$
    – Sorenai de
    Mar 24, 2022 at 10:07

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