Overview of Vector Space Models

The Need for Numerical Representation

Unlike humans, computers, inherently, do not understand text. While for us textual data is rich, complex, and highly nuanced, carrying meanings that are interpreted based on language, context, and cultural knowledge, for a computer, text is initially just a sequence of characters without inherent meaning.

To overcome these challenges, we turn to mathematical and statistical models that can process and analyze patterns within the data. However, these models require numerical input—they operate on vectors, matrices, and other mathematical structures, not on raw text.

Understanding Vector Space Models

Luckily, text representation models provide a solution for this problem, namely vector space models, which we will cover in this course.

The mathematical concept can be defined as follows. Assume we have a document D in the vector space of documents V.

The number of dimensions or columns for each document will be the total number of unique terms or words across all documents in the vector space. Therefore, the vector space can be denoted as:

where each document contains different words. Essentially, this vector space represents the vocabulary.

Now, we can represent a document in the vector space as follows:

where W_Dn denotes the weight of word n in document D. Let's take a look at an example with 2 documents and unique terms (words):

Using these vector representations we could, for example, calculate the similarity score of these documents by calculating the angle between them (cosine of the angle to more precise) to find out how similar semantically they are.

Words as Vectors

This concept, however, can be extended to individual word representations through the technique known as word embeddings. Word embeddings operate under a similar mathematical principle but focus on representing individual words as vectors rather than entire documents. The dimensions in these vectors capture latent semantic features that are not directly interpretable.

Here is an example with 2-dimensional embeddings for three words:

As you can see, words "woman" and "queen" as well as "queen" and "king" are rather similar and close to each other, while "woman" and "king" are rather far away from each other representing their semantic difference

Applications of Vector Space Models

Vector space models underpin a variety of NLP tasks, enabling:

• Semantic Similarity: Computing the similarity between text documents or words based on their vector representations;
• Information Retrieval: Enhancing search engines and recommendation systems to find content relevant to a user's query;
• Text Classification and Clustering: Automatically categorizing documents into predefined classes or grouping similar documents together;
• Natural Language Understanding: Facilitating deeper linguistic analyses that pave the way for applications like sentiment analysis, topic modeling, and more.