Language model has the *curse of dimensionality*, namely, if we want to model
the joint distribution of 10 words in a vocabulary of \(10^5\), there are
\(10^{5\times 10} = 10^{50}\) parameters. But if the variables are continuous,
we can use fewer parameters.

The goal of this paper is to find a conditional probability model. If \(w_i^j\) is the sequence of words \((w_i,w_{i+1},\dots,w_j)\), then the probability of a sentence of \(T\) is a product of conditional probabilities:

\[P(w_1^T) = \prod_{i=1}^T P(w_i\mid w_1^{i-1})\]The simplest of such is the \(n\)-gram model, which \(T=n\) considered only the combination of observed successive words. New combinations should not be assigned zero probability but difficult to find the correct value.

In this paper, the proposal is to assign each word a feature vector in
\(\mathbb{R}^m\) (e.g., \(m=30\)), then the probability function of word
sequences is same as the probability function of the feature vectors, which
both the vectors and the probability function can be *learned* simultaneously.
The learning is to maximize the log-likelihood of the training data. And the
conditional probability function in the formula above can be implemented as
a neural network of predicting the next word based on the previous ones.

Notations:

- words \(w_1,\cdots,w_T\) are from vocabulary \(V\), which \(\mid V\mid\) is large
- matrix \(C\) is a \(\mid V\mid \times m\) matrix, which row \(t\) maps word \(w_t\) to \(\mathbb{R}^m\). Denote the vector as \(C(w_t)\).
- \(n\)-gram prediction function \(g(i, C(w_{t-1}), \cdots, C(w_{t-n+1}))\) gives the probability \(P(w_t = i\mid w_{t-n+1}^{t-1})\); which can be implemented as a neural network
- log-likelihood, with regularization function \(R(\theta)\):

The function \(g(\cdot)\) is implemented as a neural network:

\[\begin{aligned} P(w_t\mid w_{t-n+1}^{t-1}) = y &= \text{softmax}(b + Wx + U\tanh(d + Hx)) \\ x &= [C(w_{t-1}), \cdots, C(w_{t-n+1})] \end{aligned}\]with:

- the word feature layer \(C\)
- a hidden layer with tanh activation
- softmax output layer

and in this network:

- the direct connection from input \(x\) to output \(y\) is optional, i.e., \(W=0\)
- the input bias \(b\) is a vector of \(\mid V\mid\) elements
- the hidden layer bias \(d\) is a vector of \(h\) elements and weight \(H\) is a \(h\times (n-1)m\) matrix, where \(h\) is the number of units in this layer, e.g. 50
- output weight \(U\) for hidden layer is a \(\mid V\mid\times h\) matrix
- output weights \(W\) for input, if not omitted, is a \(\mid V\mid\times (n-1)m\) matrix

Overall, the parameters are \(b, d, W, U, H, C\). In this model, the total number of parameters scales linearly with \(\mid V\mid\) and \(n\), hence not subject to the curse of dimensionality.

The direction connection is found to double the speed of convergence, with cost of slightly less perplexity (i.e., geometric average of \(1/P(w_t\mid w_1^{t-1})\)).

Once the weight matrix \(C\) and the network is trained, the out-of-vocabulary word can be assigned the vector

\[\sum_{i\in V} C(i)P(i\mid w_{t-n+1}^{t-1})\]The paper has a picture for the proposed neural network. The Keras code for the equivalent should be as follows:

```
from tensorflow.keras.models import Model
from tensorflow.keras.layers import Embedding, Concatenate, Dense, Input
vocab_size = 100_000 # size of vocab, after lemmatization
embed_dim = 30 # embedding vector dimension
n = 10 # n as in n-gram
hidden_dim = 50 # num of hidden units
# input is a vector of integers
inputs = Input(shape=(n,), name="input")
embedding = Embedding(input_dim=vocab_size,
output_dim=embed_dim,
mask_zero=True, # input value 0 is a mask
input_length=n,
name="embed")(inputs)
hidden = Dense(hidden_dim, activation="tanh", name="hidden")(embedding)
concat = Concatenate()([embedding, hidden])
outputs = Dense(vocab_size, activation="softmax", name="prob")(concat)
model = Model(inputs=inputs, outputs=outputs, name="NPLM")
```

## Bibliographic data

```
@article{
title = "A Neural Probabilisitc Language Model",
author = "Yoshua Bengio and Réjean Ducharme and Pascal Vincent and Christian Jauvin",
booktitle = "J. Machine Learning Research",
volume = "3",
pages = "1137--1155",
year = "2003",
}
```