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34 lines
2.4 KiB
TeX
34 lines
2.4 KiB
TeX
Vector search is for example used in Semantic Search and of course search engines.
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It is also used to find relevant data to augment an LLM prompt (Deep Research or whatever that type of feature is called).
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We have at least four metrics for the quality of results:
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\begin{itemize}
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\item \bi{Recall}: relevant items retrieved by the vector search ($R$) divided by all relevant items ($T$), so $R \div T$. The closer to $1$, the better
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\item \bi{Cosine Similarity} $S_C$: The cosine of the angle between the two vectors. Values range from $-1$ to $1$ and the closer to $1$, the better (higher = better)
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\item \bi{Cosine Distance} $D_C$: defined as $1 - S_C$, values range from $0$ to $2$ and the closer to $0$, the better (lower = better)
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\item \bi{Euclidian distance}: The norm of the vector between the two vectors. Values are from $0$ to $\8$, the lower the better.
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\end{itemize}
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$S_C(A, B) = \frac{A \cdot B}{||A|| \cdot ||B||}$, a
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\subsection{Trees/Clustering}
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Since nearest neighbour search (NNS) is not viable for large scale datasets (which they typically are),
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we use clustering algorithms such as K-Means to find $K$ cluster centroids.
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We then assign to each of the centroids the vectors that belong to this cluster
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Then, for search, we can check which cluster(s) the query vector is closest to.
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Due to the low number of centroids, this is comparatively cheap to do.
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We can then search all the vectors in the corresponding clusters, which is much cheaper due to the much lower number of vectors to search.
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Still, Vector Search is a very expensive operation due to the large number of vectors to be searched, even if the search itself is $\tco{n}$,
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with $n$ being the number of vectors to search through.
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On insert, we search the the closest centroid and simply assign the vector to that cluster.
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\subsubsection{Inverted File}
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A not (directly, as far as I can tell) covered, but very much relevant index is the IVF index (Inverted File).
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It works similarly to a normal clustered index, just that each centroid holds an \textit{inverted list} of assigned vectors.
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The \texttt{nlist} parameter defines the number of coarse clusters. The larger, the shorter the lists are, the (typically and up to a point) faster the scans are,
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but training / constructing it takes longer and it uses more RAM.
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\texttt{nprobe} sets the number of items to retrieve on each query, so for instance, setting it to one,
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we will only probe (access / retrieve) the nearest centroid's list.
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