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Understanding Documents as Vectors: A Simplified Guide

Explore the concept of documents as vectors and learn how to turn documents into vectors for efficient information retrieval. Understand the importance of cosine similarity in ranking and measuring document proximity. Dive into the query-as-vector model and its implementation in information retrieval systems. Discover the nuances of cosine similarity and the implications for document ranking.

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Understanding Documents as Vectors: A Simplified Guide

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  1. Documents as vectors • Each doc j can be viewed as a vector of tf.idf values, one component for each term • So we have a vector space • terms are axes • docs live in this space • even with stemming, may have 20,000+ dimensions • The corpus of documents gives us a matrix, which we could also view as a vector space in which words live – transposable data

  2. Why turn docs into vectors? • Query-by-example • Given a doc D, find others “like” it. • Now that D is a vector, => Given a doc, find vectors (docs) “near” it. • Intuition: t3 d2 d3 d1 θ φ t1 d5 t2 d4 Postulate: Documents that are “close together” in vector space talk about the same things.

  3. The vector space model Query as vector: • We regard query as short document • We return the documents ranked by the closeness of their vectors to the query, also represented as a vector. • Developed in the SMART system (Salton, c.1970) Standard used by TREC participants and web IR systems t3 d2 d3 d1 θ φ t1 d5 t2 d4

  4. Desiderata for Proximity • Symmetry: • If d1 is near d2, then d2 is near d1. • Triangular inequality: • If d1 near d2, and d2 near d3, then d1 is not far from d3. • Identity: • No doc is closer to dthan d itself. t3 d2 d3 d1 θ φ t1 d5 t2 d4

  5. First cut • Distance between d1 and d2 is the length of the vector |d1 – d2|. • Euclidean distance • Why is this not a great idea? • We still haven’t dealt with the issue of length normalization • Long documents would be more similar to each other by virtue of length, not topic • However, we can implicitly normalize by looking at angles instead

  6. t 3 d 2 d 1 θ t 1 t 2 Cosine similarity • Distance between vectors d1 and d2capturedby the cosine of the anglex between them. • Note – this is similarity, not distance • You do not have triangular inequality, but good enough

  7. Cosine similarity • A vector can be normalized (given a length of 1) by dividing each of its components by its length – here we use the L2 norm • This maps vectors onto the unit sphere: • Then, • Longer documents don’t get more weight

  8. Cosine similarity • Cosine of angle between two vectors • The denominator involves the lengths of the vectors. Normalization

  9. Normalized vectors • For normalized vectors, the cosine is simply the dot product:

  10. Example • Docs: Austen's Sense and Sensibility, Pride and Prejudice; Bronte's Wuthering Heights. tf weights • cos(SAS, PAP) = .996 x .993 + .087 x .120 + .017 x 0.0 = 0.999 • cos(SAS, WH) = .996 x .847 + .087 x .466 + .017 x .254 = 0.889

  11. Cosine similarity exercises • Rank the following by decreasing cosine similarity: • Two docs that have only frequent words (the, a, an, of) in common. • Two docs that have no words in common. • Two docs that have many rare words in common (wingspan, tailfin).

  12. Queries in the vector space model Central idea: the query as a vector: • We regard the query as short document • We return the documents ranked by the closeness of their vectors to the query, also represented as a vector. • Note that dq is very sparse!

  13. Summary • What’s the real point of using vector spaces?: • A user’s query can be viewed as a (very) short document. • Query becomes a vector in the same space as the docs. • Can measure each doc’s proximity to it. • Natural measure of scores/ranking – no longer Boolean. • Is it the only thing that people use? • No; if you know that some words have special meaning in your own corpus, not justified by tf.idf,then manual editing of weights may take place • E.g., “benefits” might be re-weighted by hand in the corpus of an HR department

  14. Digression: spamming indices • This was all invented before the days when people were in the business of spamming web search engines. Consider: • Indexing a sensible passive document collection vs. • An active document collection, where people (and indeed, service companies) are shaping documents in order to maximize scores • Vector space similarity may not be as useful in this context.

  15. Issues to consider • How would you augment the inverted index to support cosine ranking computations? • Walk through the steps of serving a query. • The math of the vector space model is quite straightforward, but being able to do cosine ranking efficiently at runtime is nontrivial

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