The bag-of-words model is a foundational approach for representing text documents as numerical data. In this model, you treat each document as a collection of words, disregarding grammar and word order but preserving word frequency. This abstraction reduces the rich structure of natural language into a simple vector of counts, where each dimension corresponds to a unique word in the vocabulary. By doing so, you can compare, analyze, and manipulate documents mathematically, which is essential for tasks like document classification, clustering, and similarity measurement. The main implication of this approach is that all context, syntax, and semantics are ignored — only the presence and frequency of words matter. This simplification is powerful for large-scale text mining but comes at the cost of losing nuanced meaning within the text.

To operationalize the bag-of-words model, you construct a document-term matrix. In this matrix, each row represents a document and each column corresponds to a unique term from the entire collection. The value at the intersection of a row and column is the count of how many times that term appears in the document. For instance, consider a collection of three short documents: "apple orange apple", "banana apple", and "orange banana banana". The vocabulary extracted from this collection consists of "apple", "orange", and "banana". The resulting matrix has three rows (one for each document) and three columns (one for each term). As the number of documents and unique terms increases, the matrix grows in both dimensions, often resulting in a very high-dimensional space. Most entries in this matrix are zero because any single document typically contains only a small subset of all possible terms, leading to sparsity. This means that for large collections, the document-term matrix is mostly empty, which poses both computational challenges and opportunities for optimization.

Viewing each document as a vector in a high-dimensional space provides a geometric perspective on text analysis. Each dimension of this space represents a term, and a document's vector has nonzero values only for the terms it contains. Because documents usually contain just a few terms from the entire vocabulary, these vectors are extremely sparse — most components are zero. In this space, the position of each document is determined solely by its word counts. Importantly, each dimension is independent of others, so the presence of one term does not affect the representation of another. This independence allows you to use geometric operations, such as measuring angles or distances between vectors, to quantify the similarity or difference between documents. However, the high dimensionality and sparsity also mean that typical geometric intuition may not always apply, and specialized methods are needed to work effectively with such data.