Meaning is stored as position. Position is inherited from the texts that already existed.

The vector space model represents documents and queries as numerical vectors, so that similarity becomes a geometric question. Retrieval turns into measuring distance. It is the foundation under almost everything that followed.

An algebraic model converts text into vectors, where geometric relations stand for similarity of content. The conversion is the whole trick. Everything that cannot be converted does not participate in retrieval.

A bag-of-words representation maps a document to a vector of term occurrences, discarding word order. It is crude and it works. Its limits are the subject of most of the gaps below.

Tf-idf weights a term by how often it occurs in a document and how rare it is across the collection. Common words count for little, distinctive ones for much. It is the reason a page about a narrow subject can outrank a large general one.

Cosine similarity measures the angle between two vectors rather than the distance between their tips. Direction counts and length does not, so a long document and a short one about the same subject score alike. It is the standard scoring metric.

The generalized vector space model was built to remove the assumption that terms are independent of one another. It allows terms to be related. It is the first acknowledgment that the classical model assumes something false.

SMART is the historical retrieval system in which the vector space model was first implemented. It is where the theory became an instrument. Much of the vocabulary in use comes from it.

The classical model assumes every term is independent of every other, at right angles in the space. Synonyms therefore count as unrelated. The assumption is convenient and wrong.

A document vector is the numeric representation of one document by its weighted terms. It is what gets stored and compared. Whatever is not in the vector does not exist to the system.

A query vector is the same representation applied to a search request. Retrieval compares it against every document vector. The comparison is the search.

Sparse means most entries are zero, because any document contains a small fraction of the vocabulary. Sparsity makes classical retrieval fast. It also makes the representation unable to see similarity where no words are shared.

Relevance ranking orders results by their computed similarity to the query. It is the output of everything above. The order is a consequence of the geometry rather than a judgment.

Keyword matching checks for exact string matches instead of comparing numbers. It is the approach the vector space model replaced. Writing as though it were still in force is the most common mistake in optimization advice.

The standard Boolean model retrieves documents that satisfy a logical condition, with no ordering among them. The vector space model improved on it by producing a ranking. It is why results are a list rather than a set.

It explains representation, weighting and scoring, and stops at the classical model. It leaves out every technique developed since for handling meaning rather than words, every structure that makes retrieval fast at scale, and every alternative geometry. That is roughly thirty years of the field.

A term-document matrix holds all document vectors together in one structure. Descriptions list document and query vectors separately without naming the structure they form. The matrix is what the later decomposition methods operate on.

The extended Boolean model combines logical conditions with graded matching. Sources present Boolean and vector approaches as separate alternatives. The hybrid resolves the choice and is not mentioned.

Latent semantic analysis finds hidden dimensions of meaning behind term co-occurrence, so that documents sharing no words can still be close. Texts stop at bag-of-words. This is the step where retrieval first stopped needing the exact word.

No. Latent semantic indexing is a real retrieval technique and “LSI keywords” is not a thing that search engines use. Google has stated plainly that there is no such thing, and the tools selling LSI keywords are supplying related terms with an acronym attached that has nothing to do with them. The technique and the marketing term share four letters and nothing else.

It matters historically and it is not what search engines run. It was published in 1990 and patented, and it demonstrated that documents sharing no words can be recognised as related. It does not scale to the web, because the decomposition has to be recomputed as documents arrive and the matrix at web size is unmanageable. What replaced it does the same job differently, through learned dense representations.

Singular value decomposition is the mathematical operation that extracts those hidden dimensions. It is the mechanism behind latent semantic analysis. Foundational texts present the model without the operation that repaired it.

The soft cosine measure compares vectors while allowing related terms to count as partly the same. It removes the orthogonality assumption from the scoring step. Material assuming strict orthogonality has no place for it.

Non-negative matrix factorization decomposes the matrix into parts that can be read as topics. It gives an interpretable low-rank representation. Discussions rely on tf-idf weights alone.

Random projection compresses high-dimensional vectors into fewer dimensions while approximately preserving distances. It is how large collections became tractable. Sources treat sparse representations without any compression.

The inner product is the operation underneath cosine similarity, before normalisation. Texts name the similarity and omit the operation. Knowing it explains why length normalisation is needed at all.

A dense representation encodes meaning in a few hundred continuous numbers rather than thousands of mostly-zero counts. It is what modern retrieval uses. Documentation focused on sparse tf-idf describes a previous era.

L2 normalization scales a vector to unit length, which is what makes cosine similarity independent of document length. Sources state the similarity without explaining the normalisation. It is the reason a short page can compete with a long one.

A term interaction matrix records how strongly terms relate to each other. It replaces the assumption that they do not. Texts keep the assumption instead.

The quantum vector space model applies the mathematics of quantum mechanics to represent interference and context effects between terms. It is a serious research direction. Classical presentations do not mention that alternatives exist.

A vector database stores dense vectors and retrieves the nearest ones at scale. It is the infrastructure behind current answer systems. Literature treats the historical system and skips the one in production.

Approximate nearest neighbour search finds close vectors without comparing against every candidate. Texts assume an exact scan. No system at scale does an exact scan, and the approximation shapes what is found.

The distributional hypothesis holds that words occurring in similar contexts have similar meanings. It is the linguistic claim on which every embedding rests. Sources define the methods without citing the claim that licenses them.

A word embedding places each word at a position learned from its contexts, so that related words end up close. It replaces counting with learning. The baseline account stops at discrete counts.

Latent Dirichlet allocation models documents as mixtures of topics, probabilistically rather than algebraically. Texts cover the algebraic extensions and omit the probabilistic ones. The two traditions answer the same question differently.

Principal component analysis projects data onto the directions of greatest variance. It is the statistical counterpart to the decomposition methods above. Foundational material omits it.

Locality-sensitive hashing groups similar vectors into the same buckets so that candidates can be found without full comparison. It is one of the standard speed techniques. Retrieval is presented as exact evaluation instead.

It is a graph structure that makes nearest-neighbour search fast by navigating through layers of connections. It is the index behind most current vector databases. Source material excludes graph-based indexing entirely.

Euclidean distance measures the straight-line gap between vector tips, in contrast to the angle. Discussions rely on cosine alone. Which metric is used changes which documents count as close.

The Jaccard index compares sets by their overlap rather than by weighted values. It is the set-theoretic alternative. Texts focus on real-valued weights and omit it.

Subspace clustering partitions documents by grouping them within different regions of the space. It acknowledges that one global space may not suit everything. Sources analyse the space as a single whole.

Dense retrieval encodes query and document with neural encoders and matches the resulting vectors. It is the successor to keyword-based retrieval and the basis of retrieval-augmented answering. Material focused on classical matching describes what it replaced.

A semantic vector space treats position as meaning rather than as a lexical index. Sources treat the space strictly as an index of words. The shift from words to meaning is the single largest change in the field.

Vector quantization compresses vectors by replacing them with codes from a learned codebook. It trades a little accuracy for large savings in memory. Texts discuss sparse vectors without addressing compression.

Cosine distance loss uses the similarity as a training objective rather than only as a score. It turns the metric into something a model is optimised against. Cosine is treated as a static evaluation measure.

Probabilistic latent semantic analysis models the same hidden structure statistically rather than algebraically. Algebraic extensions are covered and the statistical family is not. It is the direct predecessor of topic modelling.

A subspace embedding gives theoretical guarantees that a lower-dimensional projection preserves the relevant structure. It is why compression is safe rather than merely convenient. Literature omits the guarantees.

Information geometry treats spaces of probability distributions as curved rather than flat. Vector spaces are assumed to be flat Euclidean spaces. Whether that assumption fits meaning is an open question nobody raises.

A concept vector space represents concepts rather than text units. Sources treat the space strictly for text indexing. Answer systems increasingly reason over concepts, which is a different object from a document.

A dual vector space consists of the linear functionals acting on the original space. Vector operations are presented in the primal form only. The dual view supplies tools for reasoning about what a scoring function is.

A non-Euclidean space allows curvature, which suits hierarchical relations that flat space represents poorly. All representations here adhere to flat geometry. Hierarchies are common in knowledge and awkward in flat space.

A doxastic vector space maps belief states rather than lexical features. Texts confine vector spaces to words. Since generated answers assert beliefs, representing belief rather than vocabulary is the open frontier.

Your page occupies a position determined by what it says and what else exists nearby. If it restates what a thousand other pages say, it sits in a crowded region and is indistinguishable from its neighbours. A page carrying something nothing else carries occupies an empty region, where it is the nearest answer by default.

Write what is not already dense. The classical model rewarded distinctive terms; the modern one rewards distinctive content in a distinctive region. Both punish the same thing, which is saying what has already been said.