# continuous space language models

# Model

The neural network language model has to perform two tasks: first, project all words of the context [math]\,h_j[/math] = [math]\,w_{j-n+1}^{j-1}[/math] onto a continuous space, and second, calculate the language model probability [math]P(w_{j}=i|h_{j})[/math]. The researchers for this paper sought to find a better model for this probability than the back-off n-grams model. Their approach was to map the n-1 words sequence onto a multi-dimension continuous space using a layer of neural network followed by another layer to estimate the probabilities of all possible next words. The formulas and model goes as follows:

For some sequence of n-1 words, encode each word using 1 of K encoding, i.e. 1 where the word is indexed and zero everywhere else. Label each 1 of K encoding by [math](w_{j-n+1},\dots,w_j)[/math] for some n-1 word sequence at the j'th word in some larger context.

Let P be a projection matrix common to all n-1 words and let

[math]\,a_i=Pw_{j-n+i},i=1,\dots,n-1[/math]

Let H be the weight matrix from the projection layer to the hidden layer and the state of H would be: [math]\,h=tanh(Ha + b)[/math] where A is the concatenation of all [math]\,a_i[/math] and [math]\,b[/math] is some bias vector

Finally, the output vector would be:

[math]\,o=Vh+k[/math] where V is the weight matrix from hidden to output and k is another bias vector. [math]\,o[/math] would be a vector with same dimensions as the total vocabulary size and the probabilities can be calculated from [math]\,o[/math] by applying the softmax function.

The following figure shows the Architecture of the neural network language model. [math]\,h_j[/math] denotes the context [math]\,w_{j-n+1}^{j-1}[/math]. P is the size of one projection and H and N is the size of the second hidden and output layer, respectively. When short-lists are used the size of the output layer is much smaller than the size of the vocabulary.

In contrast to standard languaFile:Qq.pngge modeling where we want to know the probability of a word i given its context, [math]P(w_{j} = i|h_{j}) [/math], the neural network simultaneously predicts the language model probability of all words in the word list: