word2vec, sum or average word embeddings?

Question

I'm using word2vec to represent a small phrase (3 to 4 words) as a unique vector, either by adding each individual word embedding or by calculating the average of word embeddings.

From the experiments I've done I always get the same cosine similarity. I suspect it has to do with the word vectors generated by word2vec being normed to unit length (Euclidean norm) after training? or either I have a BUG in the code, or I'm missing something.

Here is the code:

import numpy as np
from nltk import PunktWordTokenizer
from gensim.models import Word2Vec
from numpy.linalg import norm
from scipy.spatial.distance import cosine

def pattern2vector(tokens, word2vec, AVG=False):
    pattern_vector = np.zeros(word2vec.layer1_size)
    n_words = 0
    if len(tokens) > 1:
        for t in tokens:
            try:
                vector = word2vec[t.strip()]
                pattern_vector = np.add(pattern_vector,vector)
                n_words += 1
            except KeyError, e:
                continue
        if AVG is True:
            pattern_vector = np.divide(pattern_vector,n_words)
    elif len(tokens) == 1:
        try:
            pattern_vector = word2vec[tokens[0].strip()]
        except KeyError:
            pass
    return pattern_vector


def main():
    print "Loading word2vec model ...\n"
    word2vecmodelpath = "/data/word2vec/vectors_200.bin"
    word2vec = Word2Vec.load_word2vec_format(word2vecmodelpath, binary=True)
    pattern_1 = 'founder and ceo'
    pattern_2 = 'co-founder and former chairman'

    tokens_1 = PunktWordTokenizer().tokenize(pattern_1)
    tokens_2 = PunktWordTokenizer().tokenize(pattern_2)
    print "vec1", tokens_1
    print "vec2", tokens_2

    p1 = pattern2vector(tokens_1, word2vec, False)
    p2 = pattern2vector(tokens_2, word2vec, False)
    print "\nSUM"
    print "dot(vec1,vec2)", np.dot(p1,p2)
    print "norm(p1)", norm(p1)
    print "norm(p2)", norm(p2)
    print "dot((norm)vec1,norm(vec2))", np.dot(norm(p1),norm(p2))
    print "cosine(vec1,vec2)",     np.divide(np.dot(p1,p2),np.dot(norm(p1),norm(p2)))
    print "\n"
    print "AVG"
    p1 = pattern2vector(tokens_1, word2vec, True)
    p2 = pattern2vector(tokens_2, word2vec, True)
    print "dot(vec1,vec2)", np.dot(p1,p2)
    print "norm(p1)", norm(p1)
    print "norm(p2)", norm(p2)
    print "dot(norm(vec1),norm(vec2))", np.dot(norm(p1),norm(p2))
    print "cosine(vec1,vec2)",     np.divide(np.dot(p1,p2),np.dot(norm(p1),norm(p2)))


if __name__ == "__main__":
    main()

and here is the output:

Loading word2vec model ...

Dimensions 200
vec1 ['founder', 'and', 'ceo']
vec2 ['co-founder', 'and', 'former', 'chairman']

SUM
dot(vec1,vec2) 5.4008677771
norm(p1) 2.19382594282
norm(p2) 2.87226958166
dot((norm)vec1,norm(vec2)) 6.30125952303
cosine(vec1,vec2) 0.857109242583


AVG
dot(vec1,vec2) 0.450072314758
norm(p1) 0.731275314273
norm(p2) 0.718067395416
dot(norm(vec1),norm(vec2)) 0.525104960252
cosine(vec1,vec2) 0.857109242583

I'm using the cosine similarity as defined here Cosine Similarity (Wikipedia). The values for the norms and dot products are indeed different.

Can anyone explain why the cosine is the same?

Thank you, David

Answer 1

Cosine measures the angle between two vectors and does not take the length of either vector into account. When you divide by the length of the phrase, you are just shortening the vector, not changing its angular position. So your results look correct to me.