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Solutions

Problem 1

For when x is a scalar or a vector of length 1:

def f(x):
    if x > 0:
        return torch.sin(x)
    else:
        return torch.cos(x)

x = torch.tensor([1.0, 0.5], requires_grad=True)

y = f(x)
print(y)
y.backward()
print(x.grad)

For when x is a vector, the conditional becomes ambiguous. To handle this, we can use the python all function. Computing the backward pass requires that the error signal be a scalar. Since there are now multiple outputs of f, we can turn y into a scalar just by summing the outputs.

def f(x):
    if all(x > 0):
        return torch.sin(x)
    else:
        return torch.cos(x)

x = torch.tensor([1.0, 0.5], requires_grad=True)

y = f(x)
print(y)
y.sum().backward()
print(x.grad)

There is one last catch to this: we are forcing the fate of the entire vector on a strong "and" condition (all items must be above 0 or they will all be considered below 0). To handle things in a more granular level, there are two different methods.

Method 1: use a for a loop

def f2(x):
    output = []
    for x_i in x:
        if x_i > 0:
            output.append(torch.sin(x_i))
        else:
            output.append(torch.cos(x_i))
    return torch.stack(output)

x = torch.tensor([1.0, -1.0], requires_grad=True)
y = f2(x)
print(y)
y.sum().backward()
print(x.grad)

Method 2: use a mask

def f3(x):
    mask = (x > 0).float()
    # alternatively, mask = torch.gt(x, 0).float()
    return mask * torch.sin(x) + (1 - mask) * torch.cos(x)

x = torch.tensor([1.0, -1.0], requires_grad=True)
y = f3(x)
print(y)
y.sum().backward()
print(x.grad)

Problem 2

def cbow(phrase):
    words = phrase.split(" ")
    embeddings = []
    for word in words:
        if word in glove.word_to_index:
            embeddings.append(glove.get_embedding(word))
    embeddings = np.stack(embeddings)
    return np.mean(embeddings, axis=0)

cbow("the dog flew over the moon").shape

# >> (100,)

def cbow_sim(phrase1, phrase2):
    vec1 = cbow(phrase1)
    vec2 = cbow(phrase2)
    return np.dot(vec1, vec2) / (np.linalg.norm(vec1) * np.linalg.norm(vec2))

cbow_sim("green apple", "green apple")
# >> 1.0

cbow_sim("green apple", "apple green")
# >> 1.0

cbow_sim("green apple", "red potato")
# >> 0.749

cbow_sim("green apple", "green alien")
# >> 0.683

cbow_sim("green apple", "blue alien")
# >> 0.5799815958114477

cbow_sim("eat an apple", "ingest an apple")
# >> 0.9304712574359718
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