BigQuery - find N nearest vectors

Question

I have a bigquery table, which has a column of the repeated data type with a 512 dimensional vector (float).

I would like to run a query which finds the N most similar vectors.

In my case similarity can be simply defined as the inner product of the target vector and each vector in the database.

I have found and run the below query, which generates this across all the combinations in the table:

#standardSQL
CREATE TABLE ml.url_cosine_similarity AS
WITH pairwise AS (
  SELECT t1.url AS id_1, t2.url AS id_2
  FROM `project.dataset.table` t1
  INNER JOIN `project.dataset.table` t2
  ON t1.url < t2.url
)
SELECT id_1, id_2, ( 
  SELECT 
    SUM(value1 * value2)/ 
    SQRT(SUM(value1 * value1))/ 
    SQRT(SUM(value2 * value2))
  FROM UNNEST(a.page_vector) value1 WITH OFFSET pos1 
  JOIN UNNEST(b.page_vector) value2 WITH OFFSET pos2 
  ON pos1 = pos2
  ) cosine_similarity
FROM pairwise t
JOIN `project.dataset.table` a ON a.url = id_1
JOIN `project.dataset.table` b ON b.url = id_2

However since I do not have a good grasp of how arrays work in bigquery, I am unsure on how to change this query to take in a target vector, and return N neighbours.

Answer 1

See simplified example - it returns top 3 nearest pairs of vectors in the table

#standardSQL
WITH `project.dataset.table` AS (
  SELECT 1 id, [1,2,3,4,5] page_vector UNION ALL
  SELECT 2, [1,3,4,5,16] UNION ALL
  SELECT 3, [2,3,4,5,6] UNION ALL
  SELECT 4, [2,4,6,8,9] UNION ALL
  SELECT 5, [1,3,4,5,16] UNION ALL
  SELECT 6, [11,12,13,14,15] 
)
SELECT a.id id1, b.id id2, ( 
  SELECT 
    SUM(value1 * value2)/ 
    SQRT(SUM(value1 * value1))/ 
    SQRT(SUM(value2 * value2))
  FROM UNNEST(a.page_vector) value1 WITH OFFSET pos1 
  JOIN UNNEST(b.page_vector) value2 WITH OFFSET pos2 
  ON pos1 = pos2
  ) cosine_similarity
FROM `project.dataset.table` a
JOIN `project.dataset.table` b
ON a.id < b.id
ORDER BY cosine_similarity DESC
LIMIT 3  

with output

Row id1 id2 cosine_similarity    
1   2   5   1.0  
2   1   4   0.9986422261219272   
3   3   4   0.9962894120648842    

If you want to output the nearest vectors (let's say two) for every vector in table - see below example

#standardSQL
WITH `project.dataset.table` AS (
  SELECT 1 id, [1,2,3,4,5] page_vector UNION ALL
  SELECT 2, [1,3,4,5,16] UNION ALL
  SELECT 3, [2,3,4,5,6] UNION ALL
  SELECT 4, [2,4,6,8,9] UNION ALL
  SELECT 5, [1,3,4,5,16] UNION ALL
  SELECT 6, [11,12,13,14,15] 
)
SELECT id, ANY_VALUE(page_vector) page_vector, 
  ARRAY_AGG(
    STRUCT(id2 AS id, page_vector2 AS page_vector, cosine_similarity AS cosine_similarity) 
    ORDER BY cosine_similarity DESC 
    LIMIT 2
  ) similar_vectors
FROM (
  SELECT a.id, a.page_vector, 
    b.id id2, b.page_vector page_vector2, ( 
    SELECT 
      SUM(value1 * value2)/ 
      SQRT(SUM(value1 * value1))/ 
      SQRT(SUM(value2 * value2))
    FROM UNNEST(a.page_vector) value1 WITH OFFSET pos1 
    JOIN UNNEST(b.page_vector) value2 WITH OFFSET pos2 
    ON pos1 = pos2
    ) cosine_similarity
  FROM `project.dataset.table` a
  JOIN `project.dataset.table` b
  ON a.id != b.id
)
GROUP BY id
ORDER BY id  

this will produce below output