Document: add doc for creating vector index

[WanYixian]

Description

See preview of vector index, Vector type and functions, create vector syntax

Related Code PR

risingwavelabs/risingwave#22432
risingwavelabs/risingwave#22022
risingwavelabs/risingwave#22808

Related doc issue

Fix #601

Checklist

• I have run the documentation build locally to verify the updates are applied correctly.
• For new pages, I have updated mint.json to include the page in the table of contents.
• All links and references have been checked and are not broken.

[wenym1]

Actually, this feature deserves a separate page, instead of as a sub-feature of create index. We can refer to the README of pgvector. cc @hzxa21 @xiangjinwu

Following are mentionable items in the page.

• new data type of Vector(n), where n is the dimension of the vector (may also update the page of data type, if there is any, and link to there)
• new vector distance operators

"<->" => ExprType::L2Distance,
"<=>" => ExprType::CosineDistance,
"<+>" => ExprType::L1Distance,
"<#>" => {
    // inner product distance
}

• example to do nearest distance vector search (<-> as distance operator, order by <distance> limit 5 as top n nearest vector)

SELECT * FROM items order by embedding <-> '[3,1,2]' limit 5;

• create vector index, as mentioned in the PR description.
• other vector functions. @xiangjinwu Can you help providing a list of newly supported vector functions? (may also update the page that shows all supported functions, and link to there)

[xiangjinwu]

vector(n) operators and functions:

• cast
  • vector(n) to varchar: assign
  • varchar to vector(n): explicit
  • vector(n) to real[]: implicit
  • int[]/numeric[]/real[]/double precision[] to vector(n): assign
• distance
  • vector(n) <-> vector(n) → double precision / l2_distance(vector(n), vector(n)) → double precision
  • vector(n) <=> vector(n) → double precision / cosine_distance(vector(n), vector(n)) → double precision
  • vector(n) <+> vector(n) → double precision / l1_distance(vector(n), vector(n)) → double precision
  • vector(n) <#> vector(n) → double precision (note this is negative inner product)
  • inner_product(vector(n), vector(n)) → double precision
• others
  • vector(n) + vector(n) → vector(n): element-wise addition
  • vector(n) - vector(n) → vector(n): element-wise subtraction
  • vector(n) * vector(n) → vector(n): element-wise multiplication
  • vector(m) || vector(n) → vector(m+n): concatenate
  • vector_norm(vector(n)) → double precision: Euclidean norm
  • l2_normalize(vector(n)) → vector(n): Normalize with Euclidean norm

[WanYixian]

Actually, this feature deserves a separate page, instead of as a sub-feature of create index. We can refer to the README of pgvector. cc @hzxa21 @xiangjinwu

Following are mentionable items in the page.

• new data type of Vector(n), where n is the dimension of the vector (may also update the page of data type, if there is any, and link to there)
• new vector distance operators

"<->" => ExprType::L2Distance,
"<=>" => ExprType::CosineDistance,
"<+>" => ExprType::L1Distance,
"<#>" => {
    // inner product distance
}

• example to do nearest distance vector search (<-> as distance operator, order by <distance> limit 5 as top n nearest vector)

SELECT * FROM items order by embedding <-> '[3,1,2]' limit 5;

• create vector index, as mentioned in the PR description.
• other vector functions. @xiangjinwu Can you help providing a list of newly supported vector functions? (may also update the page that shows all supported functions, and link to there)

@wenym1 @xiangjinwu Updated the doc accordingly! Please take another look

[wenym1]

Rest LGTM

[wenym1]

Rest LGTM. Thanks for the doc!