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One of life's most satisfying pleasures is going to a good library and
browsing in an area of interest. After having negotiated the library's
organization and found just which floor and shelves are associated with
the **CALL NUMBERS** of your topic, you are *physically*
surrounded by books and books, all of interest! Some are reassuring old
friends, already known to you; others are new books by familiar authors,
and (best of all!) some are brand new titles.

This system works because human catalogers have proven themselves able to reliably and consistently identify the (primary!) topic of books according to conventional systems of subject headings like the Library of Congress Subject Headings, or the Dewey Decimal system.

Our goal is to abstract
away from this very friendly notion of *physical* space in the
library, to a similar but generalized notion of **SEMANTIC** space in
which documents __about__ the same topic remain close together. But
rather than allowing ourselves to be restricted by the physical
realities of three-dimensional space, and the fact that books can only
be shelved in a single place in a library, we will consider abstract
spaces of thousands of dimensions. we can represent in electronic
indices.}

We can make concrete progress towards these lofty goals
beginning with the **\( \MATHNAME{INDEX} \) MATRIX** relating each
document in a corpus to all of their keywords. A very natural and
influential interpretation of this matrix (due to Gerry Salton [Salton75] [Salton83] ) is to imagine each and
every keyword of the vocabulary as a separate dimension of a **VECTOR
SPACE** . In other words, the **DIMENSIONALITY** of the vector
space is the size of our vocabulary. Each document can be represented as
a vector within such a space. Figure *(figure)* shows a very
simplified (binary) \( \mathname{Index} \) matrix, and a cartoon of its
corresponding vector representation.

Estimates of vocabulary sizes of native speakers of a language approach 50,000; if you are articulate, your speaking and/or reading vocabularies might be 100,000 or more. Assuming that we have an even modest $10^6$ document corpus, this matrix is therefore something like $10^6\times\; 10^5$. That's a big matrix, even by modern supercomputing standards.}.

In addition to the vectors
representing all documents, one special row matrix and vector
corresponds to a query. Since documents and queries exist within a
common vector space, we naturally characterize how we'd like our
retrieval system to work - just as we go to a physical location in the
library to be *near* books __about__ a topic, we seek those
documents that are *close to* our query vector. This is a useful
characterization of what we'd like our retrieval system to accomplish,
but it is still far from a specification of an algorithm for
accomplishing it. For example, it seems to require that the query vector
be compared against each and every document, something we can hope to
avoid.

An even more important issue to be resolved before the vector
space model can be useful is being specific about just what it means for
a document and query to be close to one another. As will be discussed in
Section §5.2.2 , there are many
plausible measures of proximity within a vector space. For the time
being, we will assume the use of the **INNER PRODUCT** of query and
document vectors as our metric:

People have difficulty imaging spaces
with more than the three physical dimensions of experience, and so it is
no wonder that abstract spaces of 10^5 \) dimensions are hard to
conceptualize. Sketches like *(FOAref)* do the best they can to
convey ideas in the three dimensions we appreciate, but it is critically
important that we not let intuitions based on such small-dimensional
experiences bias our understanding of the large-dimensional spaces
actually being represented and searched.

- 3.4.1 Keyword discrimination
- 3.4.2 Vector length normalization
- 3.4.3 Summary: SMART weighting specification

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