Exercise 3: Why Sparsity Is Essentially Unavoidable for Bag-of-Words Vectors — Possible Solution
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WHAT SPARSITY MEANS, PER THIS CHAPTER
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Per this chapter, "a real corpus can easily have a vocabulary of tens
of thousands of unique words — but any single document only ever uses
a tiny fraction of them. The resulting vectors are overwhelmingly
zeros, a property called sparsity." A sparse vector is one where the
vast majority of its own positions hold the value zero, with only a
small number of positions holding any nonzero value at all.

WHY THIS FOLLOWS DIRECTLY FROM HOW BAG-OF-WORDS VECTORS ARE BUILT
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Per this chapter's own definition, every vocabulary word — every
unique word that appears ANYWHERE across the entire corpus — gets its
own fixed position in every single document's own vector, regardless of
whether that particular document actually uses that word. A single
ordinary document (a short review, an email, a single news article)
realistically only contains a few dozen to a few hundred distinct
words, drawn from a vocabulary that, across an entire corpus, can
easily span tens of thousands of unique words. Every vocabulary word
NOT among those few dozen or few hundred used in this one specific
document gets a zero at its own position.

WHY THIS IS A MATTER OF SCALE, NOT A RARE COINCIDENCE
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If a corpus's own vocabulary has, say, 20,000 unique words, and a
typical document in that corpus uses only 150 distinct words, then
19,850 out of 20,000 vector positions — more than 99% — are
necessarily zero for that document, simply as an arithmetic consequence
of how few of the corpus's own total unique words any single document
could realistically contain. This isn't something that happens to occur
occasionally; it's the mathematically guaranteed outcome of building a
vector as large as the full corpus vocabulary while representing a
document that only ever uses a small subset of it.

WHY THIS IS ESSENTIALLY UNAVOIDABLE, NOT A FIXABLE DESIGN CHOICE
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The vocabulary must, by the bag-of-words method's own definition,
include every word that appears anywhere in the corpus — shrinking the
vocabulary to avoid sparsity would mean discarding legitimate words that
some OTHER document in the corpus genuinely needs represented. Given
that requirement, and given that any single document only ever uses a
small fraction of an entire corpus's own vocabulary, high sparsity is
a direct, structural consequence of the method itself — not a symptom
of a poorly-chosen or fixable parameter.

WHY THIS MATTERS PRACTICALLY, PER THIS CHAPTER
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Per this chapter, "most libraries (including CountVectorizer) store and
compute with sparse vectors efficiently rather than wastefully storing
every zero" — acknowledging sparsity as a genuine, expected property
that real tooling is specifically built to handle efficiently, rather
than an unusual failure state to be avoided.

WHY THIS WORKS AS AN ANSWER
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It explains precisely why every vocabulary word gets a position in
every document's vector, explains why any one document's own limited
word usage against a much larger corpus-wide vocabulary makes the
overwhelming majority of those positions zero as a simple matter of
scale, and explains why this is a structural consequence of the method
rather than an avoidable edge case.