# Scale cosine similarity between vectors to range 0, 1?

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

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, mathematical meaning of cosine similarity. If not, are other similarity metrics advised?

• What are you using norm_sim for? Norm_sim is monotonic in sim; is that good enough? Another option is to force both vectors to be positive so that their cosine similarity to be positive. – Angela Richardson May 27 at 7:07
• norm_sim will be used in a machine learning experiment of a colleague. The prerequisite is that all values are between 0 and 1. I am not sure how I can force vectors to be positive during training. I don't think that is possible, as far as I know. – Bram Vanroy May 27 at 7:39