Cosine similarity between 0 and 1

Question

I am interested in calculating similarity between vectors, however this similarity has to be a number between 0 and 1. There are many questions concerning tf-idf and cosine similarity, all indicating that the value lies between 0 and 1. From Wikipedia:

In the case of information retrieval, the cosine similarity of two documents will range from 0 to 1, since the term frequencies (using tf–idf weights) cannot be negative. The angle between two term frequency vectors cannot be greater than 90°.

The peculiarity is that I wish to calculate the similarity between two vectors from two different word2vec models. These models have been aligned, though, so they should in fact represent their words in the same vector space. I can calculate the similarity between a word in model_a and a word in model_b like so

import gensim as gs
from sklearn.metrics.pairwise import cosine_similarity

model_a = gs.models.KeyedVectors.load_word2vec_format(model_a_path, binary=False)
model_b = gs.models.KeyedVectors.load_word2vec_format(model_b_path, binary=False)

vector_a = model_a[word_a].reshape(1, -1)
vector_b = model_b[word_b].reshape(1, -1)

sim = cosine_similarity(vector_a, vector_b).item(0)

But sim is then a similarity metric in the [-1,1] range. Is there a scientifically sound way to map this to the [0,1] range? Intuitively I would think that something like

norm_sim = (sim + 1) / 2

is okay, but I'm not sure whether that is good practice with respect to the actual meaning of cosine similarity. If not, are other similarity metrics advised?

The reason why I am trying to get the values to be between 0 and 1 is because the data will be transferred to a colleague who will use it as a feature for her machine learning system, which expects all values to be between 0 and 1. Her intuition was to take the absolute value, but that seems to me to be a worse alternative because then you map opposites to be identical. Considering the actual meaning of cosine similarity, though, I might be wrong. So if taking the absolute value is the good approach, we can do that as well.

Answer 1

You have a fair reason to prefer 0.0-1.0 (though many learning algorithms should do just fine with a -1.0 to 1.0 range). Your norm_sim rescaling of -1.0 to 1.0 to 0.0 to 1.0 is fine, if your only purpose is to get 0.0-1.0 ranges... but of course the resulting value isn't a true cosine-similarity anymore.

It won't necessarily matter that the values aren't real full-range angles any more. (If the algorithm needed real angles, it'd work with -1.0 to 1.0.)

Using the signless absolute value would be a bad idea, as it would change the rank order of similarities – moving some results that are "natively" most-dissimilar way up.

There's been work on constraining word-vectors to have only non-negative values in dimensions, & the usual benefit is that the resulting dimensions are more likely to be individually interpretable. (See for example https://cs.cmu.edu/~bmurphy/NNSE/.) However, gensim doesn't support this variant, & only trying it could reveal whether it would be better for any particular project.

Also, there's other research that suggests usual word-vectors may not be 'balanced' around the origin (so you'll see fewer negative cosine-similiarities than would be expected from points in a random hypersphere), and that shifting them to be more balanced will usually improve them for other tasks. See: https://arxiv.org/abs/1702.01417v2