When you compare documents using similarity measures in high-dimensional spaces, you are essentially examining the geometric relationship between their vector representations. A high similarity score between two document vectors indicates that the vectors are close together in the vector space — often meaning they point in nearly the same direction. This proximity suggests that the documents share many features, such as common terms or similar term frequencies. On the other hand, a low similarity score tells you that the vectors are far apart or nearly orthogonal, reflecting that the documents have little in common according to the features captured in the vectorization process. In geometric terms, cosine similarity close to 1 means a small angle between vectors, while scores near 0 indicate large angles, and negative scores (if allowed) would mean the vectors point in opposite directions. The geometric structure of the document space is shaped by the features you extract and the way you represent each document as a vector.

Normalization and weighting play a crucial role in how these similarity scores are computed and interpreted. When you normalize document vectors — such as by scaling them to unit length — you ensure that the similarity measure focuses on the direction of the vectors rather than their magnitude. This is particularly important in text mining, where document length can vary widely, and you typically care more about the pattern of term usage than the sheer number of terms. Weighting schemes, such as TF-IDF, adjust the importance of each feature (term) in the vector, often downweighting common words and emphasizing rarer, more informative terms. The combination of normalization and weighting shapes the geometry of the document space, affecting which documents are considered similar and how clusters of similar documents emerge.

Underlying these similarity-based retrieval methods are several important assumptions. One key assumption is the independence of features: when you use a bag-of-words or TF-IDF representation, you treat each term as independent from the others, ignoring any potential semantic or syntactic relationships. This can lead to limitations, especially when documents share meaning but not vocabulary. Another assumption is that all features are on a comparable scale, or have been properly scaled through normalization or weighting. Without proper scaling, features with larger numeric ranges can dominate the similarity calculation, leading to misleading results. These assumptions simplify the computational process but also introduce potential sources of error or bias in retrieval and ranking tasks, especially as the dimensionality of the document space increases.