PCA on word2vec embeddings

Question

I am trying to reproduce the results of this paper: https://arxiv.org/pdf/1607.06520.pdf

Specifically this part:

To identify the gender subspace, we took the ten gender pair difference vectors and computed its principal components (PCs). As Figure 6 shows, there is a single direction that explains the majority of variance in these vectors. The first eigenvalue is significantly larger than the rest.

I am using the same set of word vectors as the authors (Google News Corpus, 300 dimensions), which I load into word2vec.

The 'ten gender pair difference vectors' the authors refer to are computed from the following word pairs:

I've computed the differences between each normalized vector in the following way:

model = gensim.models.KeyedVectors.load_word2vec_format('GoogleNews-vectors-
negative300.bin', binary = True)
model.init_sims()

pairs = [('she', 'he'),
('her', 'his'),
('woman', 'man'),
('Mary', 'John'),
('herself', 'himself'),
('daughter', 'son'),
('mother', 'father'),
('gal', 'guy'),
('girl', 'boy'),
('female', 'male')]

difference_matrix = np.array([model.word_vec(a[0], use_norm=True) - model.word_vec(a[1], use_norm=True) for a in pairs])

I then perform PCA on the resulting matrix, with 10 components, as per the paper:

from sklearn.decomposition import PCA
pca = PCA(n_components=10)
pca.fit(difference_matrix)

However I get very different results when I look at pca.explained_variance_ratio_ :

array([  2.83391436e-01,   2.48616155e-01,   1.90642492e-01,
         9.98411858e-02,   5.61260498e-02,   5.29706681e-02,
         2.75670634e-02,   2.21957722e-02,   1.86491774e-02,
         1.99108478e-32])

or with a chart:

The first component accounts for less than 30% of the variance when it should be above 60%!

The results I get are similar to what I get when I try to do the PCA on randomly selected vectors, so I must be doing something wrong, but I can't figure out what.

Note: I've tried without normalizing the vectors, but I get the same results.

Answer 1

They released the code for the paper on github: https://github.com/tolga-b/debiaswe

Specifically, you can see their code for creating the PCA plot in this file.

Here is the relevant snippet of code from that file:

def doPCA(pairs, embedding, num_components = 10):
    matrix = []
    for a, b in pairs:
        center = (embedding.v(a) + embedding.v(b))/2
        matrix.append(embedding.v(a) - center)
        matrix.append(embedding.v(b) - center)
    matrix = np.array(matrix)
    pca = PCA(n_components = num_components)
    pca.fit(matrix)
    # bar(range(num_components), pca.explained_variance_ratio_)
    return pca

Based on the code, looks like they are taking the difference between each word in a pair and the average vector of the pair. To me, it's not clear this is what they meant in the paper. However, I ran this code with their pairs and was able to recreate the graph from the paper: