#22. Operations on word vectors

* Cosine similarity

- 두 단어가 얼마나 similar한 지 계산하기 위해 cosine sililarity를 사용한다.

- 두 단어 벡터에 대해:

# GRADED FUNCTION: cosine_similarity

def cosine_similarity(u, v):
    """
    Cosine similarity reflects the degree of similariy between u and v
        
    Arguments:
        u -- a word vector of shape (n,)          
        v -- a word vector of shape (n,)

    Returns:
        cosine_similarity -- the cosine similarity between u and v defined by the formula above.
    """
    
    distance = 0.0
    
    # Compute the dot product between u and v (≈1 line)
    dot = np.dot(u, v)
    # Compute the L2 norm of u (≈1 line)
    norm_u = np.sqrt(np.sum(np.square(u)))
    
    # Compute the L2 norm of v (≈1 line)
    norm_v = np.sqrt(np.sum(np.square(v)))
    # Compute the cosine similarity defined by formula (1) (≈1 line)
    cosine_similarity = dot / (norm_u * norm_v)
    
    return cosine_similarity

 

* Word analogy task

- "A is to B as C is to _"에서 _가 무엇인지를 구해야 한다.

- ea, eb, ec, ed -> eb - ea ≈ ed - ec와 같은 관계를 가지게 됨.

# GRADED FUNCTION: complete_analogy

def complete_analogy(word_a, word_b, word_c, word_to_vec_map):
    """
    Performs the word analogy task as explained above: a is to b as c is to ____. 
    
    Arguments:
    word_a -- a word, string
    word_b -- a word, string
    word_c -- a word, string
    word_to_vec_map -- dictionary that maps words to their corresponding vectors. 
    
    Returns:
    best_word --  the word such that v_b - v_a is close to v_best_word - v_c, as measured by cosine similarity
    """
    
    # convert words to lower case
    word_a, word_b, word_c = word_a.lower(), word_b.lower(), word_c.lower()
    
    # Get the word embeddings e_a, e_b and e_c (≈1-3 lines)
    e_a, e_b, e_c = word_to_vec_map[word_a], word_to_vec_map[word_b], word_to_vec_map[word_c]
    
    words = word_to_vec_map.keys()
    max_cosine_sim = -100              # Initialize max_cosine_sim to a large negative number
    best_word = None                   # Initialize best_word with None, it will help keep track of the word to output

    # loop over the whole word vector set
    for w in words:        
        # to avoid best_word being one of the input words, pass on them.
        if w in [word_a, word_b, word_c] :
            continue

        # Compute cosine similarity between the vector (e_b - e_a) and the vector ((w's vector representation) - e_c)  (≈1 line)
        cosine_sim = cosine_similarity(np.subtract(e_b, e_a), np.subtract(word_to_vec_map[w], e_c))
        
        # If the cosine_sim is more than the max_cosine_sim seen so far,
            # then: set the new max_cosine_sim to the current cosine_sim and the best_word to the current word (≈3 lines)
        if cosine_sim > max_cosine_sim:
            max_cosine_sim = cosine_sim
            best_word = w
        
    return best_word

 

 

* Debiasing word vectors

- word embedding에 gender biases가 포함되어 있는 경우들이 존재하기 때문에 이러한 biases를 제거해주는 작업이 필요하다. 

- g = e(woman) - e(man)이라고 하자. 그러면 벡터 g는 gender에 대한 인코딩일 것이다. 

- g를 이용해 단어들간의 similarity를 계산해보면:

print('Other words and their similarities:')
word_list = ['lipstick', 'guns', 'science', 'arts', 'literature', 'warrior','doctor', 'tree', 'receptionist', 
             'technology',  'fashion', 'teacher', 'engineer', 'pilot', 'computer', 'singer']
for w in word_list:
    print (w, cosine_similarity(word_to_vec_map[w], g))

Other words and their similarities:
lipstick 0.276919162564
guns -0.18884855679
science -0.0608290654093
arts 0.00818931238588
literature 0.0647250443346
warrior -0.209201646411
doctor 0.118952894109
tree -0.0708939917548
receptionist 0.330779417506
technology -0.131937324476
fashion 0.0356389462577
teacher 0.179209234318
engineer -0.0803928049452
pilot 0.00107644989919
computer -0.103303588739
singer 0.185005181365

이런 식으로 gender stereotype을 담은 결과값들이 나온다.

- 이러한 bias를 제거해 주는 알고리즘을 사용해보자!

 

(1) Neutralize bias for non-gender specific words

- 50차원의 word embedding을 사용하고 있다고 하면 space는 다음과 같이 나눠지게 될 것이다.: the bias-direction g와 나머지 49 dimensions(g⊥)

- input embedding e에 대해 e^debiased를 구하는 식은 다음과 같다:

def neutralize(word, g, word_to_vec_map):
    """
    Removes the bias of "word" by projecting it on the space orthogonal to the bias axis. 
    This function ensures that gender neutral words are zero in the gender subspace.
    
    Arguments:
        word -- string indicating the word to debias
        g -- numpy-array of shape (50,), corresponding to the bias axis (such as gender)
        word_to_vec_map -- dictionary mapping words to their corresponding vectors.
    
    Returns:
        e_debiased -- neutralized word vector representation of the input "word"
    """
    
    ### START CODE HERE ###
    # Select word vector representation of "word". Use word_to_vec_map. (≈ 1 line)
    e = word_to_vec_map[word]
    
    # Compute e_biascomponent using the formula give above. (≈ 1 line)
    e_biascomponent = (np.dot(e, g) / (np.sum(np.square(g)))) * g
 
    # Neutralize e by substracting e_biascomponent from it 
    # e_debiased should be equal to its orthogonal projection. (≈ 1 line)
    e_debiased = e - e_biascomponent
    ### END CODE HERE ###
    
    return e_debiased

 

(2) Equalization algorithm for gender-specific words

- actress/actor 같은 경우 동일한 값을 가지도록 만들어주어야 함.

    - 예를 들어 'babysit'의 경우, actor보다 actress와의 거리가 더 가깝다. 이런 경우에도 차이가 나지 않도록 만들어 주어야 함.

def equalize(pair, bias_axis, word_to_vec_map):
    """
    Debias gender specific words by following the equalize method described in the figure above.
    
    Arguments:
    pair -- pair of strings of gender specific words to debias, e.g. ("actress", "actor") 
    bias_axis -- numpy-array of shape (50,), vector corresponding to the bias axis, e.g. gender
    word_to_vec_map -- dictionary mapping words to their corresponding vectors
    
    Returns
    e_1 -- word vector corresponding to the first word
    e_2 -- word vector corresponding to the second word
    """
    
    ### START CODE HERE ###
    # Step 1: Select word vector representation of "word". Use word_to_vec_map. (≈ 2 lines)
    w1, w2 = pair
    e_w1, e_w2 = word_to_vec_map[w1], word_to_vec_map[w2]
    
    # Step 2: Compute the mean of e_w1 and e_w2 (≈ 1 line)
    mu = (e_w1 + e_w2) / 2

    # Step 3: Compute the projections of mu over the bias axis and the orthogonal axis (≈ 2 lines)
    mu_B = np.dot(mu, bias_axis) / (np.linalg.norm(bias_axis)**2) * bias_axis
    mu_orth = mu - mu_B

    # Step 4: Use equations (7) and (8) to compute e_w1B and e_w2B (≈2 lines)
    e_w1B = np.dot(e_w1, bias_axis) / (np.linalg.norm(bias_axis)**2) * bias_axis
    e_w2B = np.dot(e_w2, bias_axis) / np.sum(np.square(bias_axis)) * bias_axis
        
    # Step 5: Adjust the Bias part of e_w1B and e_w2B using the formulas (9) and (10) given above (≈2 lines)
    corrected_e_w1B = np.sqrt((1-(np.linalg.norm(mu_orth)))**2) * (e_w1B - mu_B) / (np.abs((e_w1 - mu_orth) - mu_B))
    corrected_e_w2B = np.sqrt((1-(np.linalg.norm(mu_orth)))**2) * (e_w2B - mu_B) / (np.abs((e_w2 - mu_orth) - mu_B))

    # Step 6: Debias by equalizing e1 and e2 to the sum of their corrected projections (≈2 lines)
    e1 = corrected_e_w1B + mu_orth
    e2 = corrected_e_w2B + mu_orth
                                                                
    ### END CODE HERE ###
    
    return e1, e2

cosine similarities before equalizing:
cosine_similarity(word_to_vec_map["man"], gender) =  -0.117110957653
cosine_similarity(word_to_vec_map["woman"], gender) =  0.356666188463

cosine similarities after equalizing:
cosine_similarity(e1, gender) =  -0.710920529702
cosine_similarity(e2, gender) =  0.738567730352