How is the training comlexity of NNLM word2vec calculated?

Question

I was reading this paper on word2vec, and came around the following description of a feedforward NNLM:

It consists of input, projection, hidden and output layers. At the input layer, N previous words are encoded using 1-of-V coding, where V is size of the vocabulary. The input layer is then projected to a projection layer P that has dimensionality N × D, using a shared projection matrix. As only N inputs are active at any given time, composition of the projection layer is a relatively cheap operation.

The following expression is given for the computational complexity per training example:

Q = N×D + N×D×H + H×V.

The last two terms make sense to me: N×D×H is roughly the amount of parameters in a dense layer from the N×D-dimensional projection layer to the H hidden neurons, analogous for H×V. The first term, however, I expected to be V×D since the mapping from a one-hot encoded word to a D-dimensional vector is done via a V×D dimensional matrix. I came to that conclusion after reading this referenced paper and this SO post where the workings of the projection layer are explained in more detail.

Perhaps I have misunderstood what is meant by "training complexity".

Answer 1

Yes. Technically your understanding is correct; i.e. if all input neurons are active, the computational complexity would be like you said: Q = V$\times$D + N$\times$D$\times$H + H$\times$V.

So, if there are V words, the input, of course, will be a 1 of V one-hot vector. And since there are N words in the input, the input matrix will be of size N$\times$V. If D is the dimensionality of the embedded vector, the projection matrix will be of size V$\times$D. The product of the input matrix and the projection matrix will then have dimensions (N$\times$V)$\cdot$(V$\times$D) = N$\times$D.

But remember that this is a time series problem and not all input words are available at any given time. For a N-gram model, only N inputs are active at any given time. Hence, of the total V matrix elements in the projection matrix, only N elements corresponding to the input one-hot vector are referenced/updated at any given time. The remaining elements are referenced/updated at a different time.

As a result, at any given time, the input matrix will have dimension N$\times$N, and the projection matrix will only have to be N$\times$D. The product of these two matrices (N$\times$N)$\cdot$(N$\times$D) will still have dimensions N$\times$D. Hence the total complexity is Q = N$\times$D + N$\times$D$\times$H + H$\times$V.

But in practice, this is implemented as a look up table rather than a matrix multiplication, as mentioned here.