Keras embedding layers: how do they work?

Suppose you have N objects that do not directly have a mathematical representation. For example words.

As neural networks are only able to work with tensors you should look for some way to translate those objects to tensors. The solution is in a giant matrix (embedding matrix) where it relates each index of an object with its translation to tensor.

object_index_1: vector_1
object_index_1: vector_2
...
object_index_n: vector_n

Selecting the vector of a specific object can be translated to a matrix product in the following way:

Where v is the one-hot vector that determines which word need to be translated. And M is the embedding matrix.

If we propose the usual pipeline, it would be the following:

1. We have a list of objects.

objects = ['cat', 'dog', 'snake', 'dog', 'mouse', 'cat', 'dog', 'snake', 'dog']

2. We transform these objects into indices (we calculate the unique objects).

unique = ['cat', 'dog', 'snake', 'mouse'] # list(set(objects))
objects_index = [0, 1, 2, 1, 3, 0, 1, 2, 1] #map(unique.index, objects)

3. We transform these indices to a one hot vector (remember that there is only one where the index is)

objects_one_hot = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], 
     [0, 0 , 0, 1], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0]] # map(lambda x: [int(i==x) for i in range(len(unique))], objects_index)
#objects_one_hot is matrix is 4x9

4. We create or use the embedding matrix:

#M = matrix of dim x 4 (where dim is the number of dimensions you want the vectors to have). 
#In this case dim=2
M = np.array([[1, 1], [1, 2], [2, 2], [3,3]]).T # or... np.random.rand(2, 4)
#objects_vectors = M * objects_one_hot
objects_vectors = [[1, 1], [1, 2], [2, 2], [1, 2], 
    [3, 3], [1, 1], [1, 2], [2,2], [1, 2]] # M.dot(np.array(objects_one_hot).T)

Normally the embedding matrix is ​​learned during the same model learning, to adapt the best vectors for each object. We already have the mathematical representation of the objects!

As you have seen we have used one hot and later a matrix product. What you really do is take the column of M that represents that word.

During the learning this M will be adapted to improve the representation of the object and as a consequence the loss goes down.

The Embedding layer in Keras (also in general) is a way to create dense word encoding. You should think of it as a matrix multiply by One-hot-encoding (OHE) matrix, or simply as a linear layer over OHE matrix.

It is used always as a layer attached directly to the input.

Sparse and dense word encoding denote the encoding effectiveness.

One-hot-encoding (OHE) model is sparse word encoding model. For example if we have 1000 input activations, there will be 1000 OHE vectors for each input feature.

Let's say we know some input activations are dependent, and we have 64 latent features. We would have this embedding:

e = Embedding(1000, 64, input_length=50)

1000 tells we plan to encode 1000 words in total. 64 tells we use 64 dimensional vector space. 50 tells input documents have 50 words each.

Embedding layers will fill up randomly with non-zero values and the parameters need to be learned.

There are other parameters when creating the Embedding layer in here

What is the output from the Embedding layer?

The output of the Embedding layer is a 2D-vector with one embedding for each word in the input sequence of words (input document).

NOTE: If you wish to connect a Dense layer directly to an Embedding layer, you must first flatten the 2D output matrix to a 1D vector using the Flatten layer.

As one may easily notice - multiplication of a one-hot vector with an Embedding matrix could be effectively performed in a constant time as it might be understood as a matrix slicing. And this exactly what an Embedding layer does during computations. It simply selects an appropriate index using a gather backend function. This means that your understanding of an Embedding layer is correct.