# 4.1 Vector Space Model (VSM)

# Vector Space Model (VSM)

citation : Pro Deep Learning with Tensorflow

In NLP information-retrieval systems, a document is generally represented as simply a vector of the count of the words it contains. For retrieving documents similar to a specific document either the cosine of the angle or the dot product between the document and other documents is computed. The cosine of the angle between two vectors gives a similarity measure based on the similarity between their vector compositions. To illustrate this fact, let us look at two vectors x, y \(\in\)

[\begin{align*}
& \phi(x,y) = \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right)
= \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j) =
& (x_1, \ldots, x_n) \left( \begin{array}{ccc}
\phi(e_1, e_1) & \cdots & \phi(e_1, e_n)
\vdots & \ddots & \vdots
\phi(e_n, e_1) & \cdots & \phi(e_n, e_n)
\end{array} \right)
\left( \begin{array}{c}
y_1
\vdots
y_n
\end{array} \right)
\end{align*}]