PostgreSQL 向量

余弦相似度与余弦距离

什么是余弦相似度

余弦相似度用于衡量两个向量之间的相似程度。具体来说：

余弦相似度为 1：表示两个向量完全相同。
余弦相似度为 0：表示两个向量正交（没有相似性）。
余弦相似度为 -1：表示两个向量完全相反。

注意：在 pgvector 扩展中，运算符 <=> 实际上计算的是余弦距离，定义为
[ \text{cosine_distance} = 1 - \text{cosine_similarity} ] 因此：

当余弦相似度为 1 时，余弦距离为 0；
当余弦相似度为 0 时，余弦距离为 1；
当余弦相似度为 -1 时，余弦距离为 2。

余弦相似度公式

余弦相似度的数学定义为： [ \text{cosine_similarity} = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} ] 其中：

(\vec{A} \cdot \vec{B}) 为向量点积
(|\vec{A}|) 和 (|\vec{B}|) 为向量的范数

使用 PostgreSQL 和 `pgvector` 扩展来计算向量相似度

在 PostgreSQL 中，使用 pgvector 扩展可以方便地存储向量并计算向量之间的相似度或距离。

1. 安装 `pgvector` 扩展

确保已安装该扩展：

CREATE EXTENSION IF NOT EXISTS vector;

-- 欧氏距离 0, 两个向量完全相同，距离自是 0
SELECT '[1, 2, 3]' <=> '[1, 2, 3]' AS similarity;
-- 两个向量的内积 -14
SELECT '[1, 2, 3]' <#> '[1, 2, 3]' AS similarity;

2. 创建表并存储向量

下面创建一个示例表，假设向量维度为 3：

CREATE TABLE embeddings (
    id serial PRIMARY KEY,
    embedding vector(3)
);

插入示例数据：

INSERT INTO embeddings (embedding) VALUES
    ('[1, 2, 3]'),
    ('[4, 5, 6]');

3. 计算向量距离与相似度

（1）使用 <=> 计算余弦距离

下面查询计算向量 [1, 2, 3] 与表中向量之间的余弦距离（注意：值越小表示越相似）：

SELECT
    id,
    embedding,
    (embedding <=> '[1, 2, 3]') AS cosine_distance
FROM
    embeddings
ORDER BY
    cosine_distance ASC;

（2）计算余弦相似度

如果需要得到余弦相似度，则可用公式 cosine_similarity = 1 - cosine_distance：

WITH input_vector AS (
    SELECT '[1, 2, 3]'::vector(3) AS embedding
)
SELECT
    e.id,
    e.embedding,
    (1 - (e.embedding <=> iv.embedding)) AS cosine_similarity
FROM
    embeddings e,
    input_vector iv
ORDER BY
    cosine_similarity DESC;

在这个示例中：

e.embedding <=> iv.embedding 得到的是余弦距离；
1 - (e.embedding <=> iv.embedding) 则转换为余弦相似度（值越大越相似）。

professors 向量化和查询示例

假设有一张教授信息表，需要为姓名、描述和备注添加向量字段（维度 1536），并进行相似度查询。

1. 修改表结构添加向量字段

CREATE TABLE rumi_sjsu_professors(
  id BIGINT NOT NULL,
  name VARCHAR(256),
  department VARCHAR(256),
  job_title VARCHAR(256),
  email VARCHAR(256),
  description TEXT,
  files JSON,
  remark VARCHAR(256),
  name_vector VECTOR(1536),
  description_vector VECTOR(1536),
  remark_vector VECTOR(1536),
  vectors_completed BOOLEAN DEFAULT FALSE,
  creator VARCHAR(64) DEFAULT '',
  create_time TIMESTAMP WITH TIME ZONE NOT NULL DEFAULT CURRENT_TIMESTAMP,
  updater VARCHAR(64) DEFAULT '',
  update_time TIMESTAMP WITH TIME ZONE NOT NULL DEFAULT CURRENT_TIMESTAMP,
  deleted SMALLINT DEFAULT 0,
  tenant_id BIGINT NOT NULL DEFAULT 0,
  PRIMARY KEY (id)
);

2. 数据插入示例

下面是插入包含向量数据的示例语句：

-- 示例1
INSERT INTO rumi_sjsu_professors (
  id, name, department, job_title, email, description, files, remark, name_vector, description_vector, remark_vector, vectors_completed
) VALUES (
  1,
  'Dr. John Doe',
  'Computer Science',
  'Professor',
  '[email protected]',
  'Expert in artificial intelligence and machine learning.',
  '{"resume": "link_to_resume.pdf"}',
  'Well-known researcher',
  '[0.1, 0.2, 0.3, ...]',  -- 1536维向量
  '[0.1, 0.2, 0.3, ...]',  -- 1536维向量
  '[0.1, 0.2, 0.3, ...]',  -- 1536维向量
  TRUE
);

-- 示例2
INSERT INTO rumi_sjsu_professors (
  id, name, department, job_title, email, description, files, remark, name_vector, description_vector, remark_vector, vectors_completed
) VALUES (
  2,
  'Dr. Jane Smith',
  'Mathematics',
  'Associate Professor',
  '[email protected]',
  'Specializes in algebraic topology and number theory.',
  '{"cv": "link_to_cv.pdf"}',
  'Renowned academic',
  '[0.1, 0.2, 0.3, ...]',
  '[0.1, 0.2, 0.3, ...]',
  '[0.1, 0.2, 0.3, ...]',
  TRUE
);

-- 示例3
INSERT INTO rumi_sjsu_professors (
  id, name, department, job_title, email, description, files, remark, name_vector, description_vector, remark_vector, vectors_completed
) VALUES (
  3,
  'Dr. Emily Zhang',
  'Physics',
  'Assistant Professor',
  '[email protected]',
  'Researcher in quantum mechanics and particle physics.',
  '{"portfolio": "link_to_portfolio.pdf"}',
  'Emerging scientist',
  '[0.1, 0.2, 0.3, ...]',
  '[0.1, 0.2, 0.3, ...]',
  '[0.1, 0.2, 0.3, ...]',
  TRUE
);

3. 计算相似度的 SQL 查询

假设我们希望计算输入向量与三个向量字段的余弦相似度（需转换为 1 - (<=>)）并检索相似度最高的三条记录，可按如下方式查询：

SELECT
  id,
  name,
  department,
  job_title,
  email,
  description,
  remark,
  (1 - (name_vector <=> input_vector)) AS name_similarity,
  (1 - (description_vector <=> input_vector)) AS description_similarity,
  (1 - (remark_vector <=> input_vector)) AS remark_similarity
FROM
  rumi_sjsu_professors,
  LATERAL (
    VALUES (
      ARRAY[0.1, 0.2, 0.3, ...]::VECTOR(1536)  -- 替换为实际输入向量
    )
  ) AS input(input_vector)
ORDER BY
  GREATEST(
    1 - (name_vector <=> input_vector),
    1 - (description_vector <=> input_vector),
    1 - (remark_vector <=> input_vector)
  ) DESC
LIMIT 3;

说明：

在 SELECT 子句中，我们分别计算三个字段的余弦相似度；
ORDER BY 中使用了 GREATEST(...) DESC，确保返回的记录中至少有一个字段的相似度最高；
如果需要在 WHERE 子句中过滤出相似度较高的记录，不能直接引用 SELECT 的别名，而需要重新计算。例如：

SELECT
  id,
  name,
  department,
  job_title,
  email,
  description,
  remark,
  (1 - (name_vector <=> input_vector)) AS name_similarity,
  (1 - (description_vector <=> input_vector)) AS description_similarity,
  (1 - (remark_vector <=> input_vector)) AS remark_similarity
FROM
  rumi_sjsu_professors,
  LATERAL (
    VALUES (
      ARRAY[0.1, 0.2, 0.3, ...]::VECTOR(1536)  -- 替换为实际输入向量
    )
  ) AS input(input_vector)
WHERE
  (1 - (name_vector <=> input_vector)) > 0.9
  OR (1 - (description_vector <=> input_vector)) > 0.9
  OR (1 - (remark_vector <=> input_vector)) > 0.9
ORDER BY
  GREATEST(
    1 - (name_vector <=> input_vector),
    1 - (description_vector <=> input_vector),
    1 - (remark_vector <=> input_vector)
  ) DESC
LIMIT 3;

4. <#> and <=>

在 pgvector 扩展中，提供了不同的距离（或相似度）运算符，它们对应不同的度量方式。具体来说：

<#> 运算符（余弦距离）
- 计算内容： 它计算的是余弦距离，定义为 [ \text{cosine_distance}(v,w)=1-\frac{v\cdot w}{|v||w|} ] 也就是 1 减去余弦相似度。
- 取值范围： 余弦相似度的取值范围为 ([-1,1])，因此余弦距离的范围为 ([0,2])
  - 当两个向量方向完全一致时，余弦相似度为 1，对应余弦距离 0；
  - 当两个向量方向完全相反时，余弦相似度为 –1，对应余弦距离 2。
- 读法： 读作“余弦距离”。
<=> 运算符（内积距离）
- 计算内容： 它计算的是内积距离，通常定义为负的内积： [ \text{inner_product_distance}(v,w) = - (v\cdot w) ] 这样设计的目的是使得相似（内积较大）的向量其“距离”较小。
- 取值范围： 内积本身没有固定的上下界（除非向量做了归一化处理），因此：
  - 若向量未经归一化，其内积可能为任意实数，所以内积距离理论上取值范围为 ((-\infty,,+\infty))；
  - 若向量都做了单位归一化，那么内积的范围就是 ([-1,1])，对应的内积距离范围为 ([-1,1])。
- 读法： 读作“内积距离”。

总结一下：

<#>： 计算余弦距离，范围 [0, 2]，读作“余弦距离”。
<=>： 计算内积距离（即负内积），范围取决于向量是否归一化，未归一化时理论上无界，读作“内积距离”。

使用代码操作向量数据库

1. 使用代码插入数据

如何使用代码代替上面的 INSERT INTO rumi_sjsu_professors 语句

import java.sql.SQLException;
import java.util.Arrays;

import org.junit.Test;
import org.postgresql.util.PGobject;

import nexus.io.jfinal.plugin.activerecord.Db;
import nexus.io.jfinal.plugin.activerecord.Row;
import nexus.io.open.chat.config.DbConfig;
import nexus.io.openai.client.OpenAiClient;
import nexus.io.openai.consts.OpenAiModels;
import nexus.io.openai.embedding.EmbeddingRequestVo;
import nexus.io.openai.embedding.EmbeddingResponseVo;
import nexus.io.tio.utils.environment.EnvUtils;

import lombok.extern.slf4j.Slf4j;

@Slf4j
public class SjsuProfessorsTest {

  @Test
  public void testInsertDrEmilyZhang() {
    EnvUtils.load();
    new DbConfig().config();

    int id = 3;
    String name = "Dr. Emily Zhang";
    String description = "Researcher in quantum mechanics and particle physics.";
    String remark = "Emerging scientist";

    Row row = new Row();
    row.set("id", id).set("name", name).set("department", "Physics").set("job_title", "Associate Professor")
        .set("email", "[email protected]").set("description", description).set("remark", remark);

    setNameVector(name, row);
    setDescriptionVector(description, row);
    setRemarkVector(remark, row);

    row.setTableName("rumi_sjsu_professors");
    boolean save = Db.save(row);
    if (save) {
      log.info("success");
    }

  }

  @Test
  public void testInsertDrJaneSmith() {
    EnvUtils.load();
    new DbConfig().config();

    int id = 2;
    String name = "Dr. Jane Smith";
    String description = "Specializes in algebraic topology and number theory.";
    String remark = "Renowned academic";

    Row row = new Row();
    row.set("id", id).set("name", name).set("department", "Mathematics").set("job_title", "Associate Professor")
        .set("email", "[email protected]").set("description", description).set("remark", remark);

    setNameVector(name, row);
    setDescriptionVector(description, row);
    setRemarkVector(remark, row);

    row.setTableName("rumi_sjsu_professors");
    boolean save = Db.save(row);
    if (save) {
      log.info("success");
    }

  }

  @Test
  public void testInsertByRecord() {
    EnvUtils.load();
    new DbConfig().config();

    String name = "Dr. John Doe";
    String description = "Expert in artificial intelligence and machine learning.";
    String remark = "Well-known researcher";

    Row row = new Row();
    row.set("id", 1).set("name", name).set("department", "Computer Science").set("job_title", "Professor")
        .set("email", "[email protected]").set("description", description).set("remark", remark);

    setNameVector(name, row);
    setDescriptionVector(description, row);
    setRemarkVector(remark, row);

    row.setTableName("rumi_sjsu_professors");
    boolean save = Db.save(row);
    if (save) {
      log.info("success");
    }
  }

  private void setRemarkVector(String description, Row row) {
    PGobject nameVector = getPgVector(description);
    row.set("remark_vector", nameVector);
  }

  private void setDescriptionVector(String description, Row row) {
    PGobject nameVector = getPgVector(description);
    row.set("description_vector", nameVector);
  }

  private void setNameVector(String name, Row row) {
    PGobject nameVector = getPgVector(name);
    row.set("name_vector", nameVector);
  }

  private PGobject getPgVector(String description) {
    EmbeddingRequestVo reoVo = new EmbeddingRequestVo(description, OpenAiModels.text_embedding_3_small);
    EmbeddingResponseVo respVo = OpenAiClient.embeddings(reoVo);
    Float[] embedding = respVo.getData().get(0).getEmbedding();
    String vectorString = Arrays.toString(embedding);

    // 使用PGobject来设置vector类型
    PGobject nameVector = new PGobject();
    nameVector.setType("vector");
    try {
      nameVector.setValue(vectorString);
    } catch (SQLException e) {
      e.printStackTrace();
    }
    return nameVector;
  }
}

插入其他的数据示例省略

2.查询数据

输出结果如下,为了方便展示,我省略了 vector 的 value 部分

import org.junit.Test;

import nexus.io.jfinal.plugin.activerecord.Db;
import nexus.io.jfinal.plugin.activerecord.Row;
import nexus.io.open.chat.config.DbConfig;
import nexus.io.tio.utils.environment.EnvUtils;
import nexus.io.tio.utils.json.JsonUtils;

public class SjsuProfessorsQueryTest {

  @Test
  public void testQuery() {
    EnvUtils.load();
    new DbConfig().config();
    int id = 3;
    Row row = Db.findById("rumi_sjsu_professors", "id", id);
    String json = JsonUtils.toJson(row.toMap());
    System.out.println(json);
  }

}

{
  "id": 3,
  "name": "Dr. Emily Zhang",
  "department": "Physics",
  "job_title": "Associate Professor",
  "email": "[email protected]",
  "description": "Researcher in quantum mechanics and particle physics.",
  "remark": "Emerging scientist",
  "name_vector": [],
  "description_vector": [],
  "remark_vector": {
    "value": "[]",
    "type": "vector",
    "null": true
  },
  "vectors_completed": false,
  "creator": "",
  "create_time": "2024-07-03 23:29:44",
  "updater": "",
  "update_time": "2024-07-03 23:29:44",
  "deleted": 0,
  "tenant_id": 0
}

3. Java 向量查询

在 Java 中使用余弦相似度进行查询，并将输入向量传递给 SQL 查询.

确保 PostgreSQL 数据库已配置好 pgvector 扩展：

CREATE EXTENSION IF NOT EXISTS vector;

创建包含向量的表：

CREATE TABLE vector_test (
   id SERIAL PRIMARY KEY,
   name TEXT,
   embedding VECTOR(1536)
);

在 Java 中构建查询并传递向量参数：

可以使用 PreparedStatement 来构建和执行查询。以下是一个示例代码，展示如何将向量传递给 SQL 查询并执行查询。

package nexus.io.open.chat.model;

import java.sql.Connection;
import java.sql.DriverManager;
import java.sql.PreparedStatement;
import java.sql.ResultSet;
import java.sql.SQLException;
import java.util.Arrays;

import nexus.io.open.chat.config.DbConfig;
import nexus.io.openai.client.OpenAiClient;
import nexus.io.openai.consts.OpenAiModels;
import nexus.io.openai.embedding.EmbeddingRequestVo;
import nexus.io.openai.embedding.EmbeddingResponseVo;
import nexus.io.tio.utils.dsn.DbDSNParser;
import nexus.io.tio.utils.dsn.JdbcInfo;
import nexus.io.tio.utils.environment.EnvUtils;

public class VectorSearchTest {

  public static void main(String[] args) throws SQLException {
    // 示例输入
    EnvUtils.load();
    new DbConfig().config();
    String question = "Jane Smith";
    EmbeddingRequestVo reoVo = new EmbeddingRequestVo(question, OpenAiModels.text_embedding_3_small);
    EmbeddingResponseVo respVo = OpenAiClient.embeddings(reoVo);
    Float[] embedding = respVo.getData().get(0).getEmbedding();

    // 转换嵌入向量为 SQL 可用的字符串
    String vectorString = Arrays.toString(embedding);

    String dsn = EnvUtils.get("DATABASE_DSN");

    JdbcInfo jdbc = new DbDSNParser().parse(dsn);

    // PostgreSQL 连接配置
    String url = jdbc.getUrl();
    String user = jdbc.getUser();
    String password = jdbc.getPswd();

    // SQL 查询
    String sql = "SELECT * FROM vector_test ORDER BY embedding <=> ? LIMIT 10";

    try (Connection conn = DriverManager.getConnection(url, user, password);
        PreparedStatement pstmt = conn.prepareStatement(sql)) {

      // 设置向量参数
      pstmt.setObject(1, vectorString, java.sql.Types.OTHER);

      // 执行查询
      try (ResultSet rs = pstmt.executeQuery()) {
        while (rs.next()) {
          // 处理查询结果
          int id = rs.getInt("id");
          String name = rs.getString("name");
          // 继续处理其他字段
          System.out.println("ID: " + id + ", Name: " + name);
        }
      }
    }
  }
}

解释：

获取嵌入向量：
- 调用 OpenAiClient.embeddings 函数来获取输入问题的嵌入向量。
转换嵌入向量：
- 将 Java 中的数组转换为 PostgreSQL 可用的 [] 格式字符串。
建立数据库连接和执行查询：
- 使用 DriverManager 来建立数据库连接。
- 使用 PreparedStatement 来构建和执行查询，并传递向量参数。
- 处理查询结果。

4. 使用 Db 工具类进行向量查询

package nexus.io.open.chat.model;

import java.util.Arrays;
import java.util.List;

import org.junit.Test;

import nexus.io.jfinal.plugin.activerecord.Db;
import nexus.io.jfinal.plugin.activerecord.Row;
import nexus.io.jfinal.plugin.template.SqlTemplates;
import nexus.io.open.chat.config.SqlTplsConfig;
import nexus.io.open.chat.config.DbConfig;
import nexus.io.openai.client.OpenAiClient;
import nexus.io.openai.consts.OpenAiModels;
import nexus.io.openai.embedding.EmbeddingRequestVo;
import nexus.io.openai.embedding.EmbeddingResponseVo;
import nexus.io.tio.utils.environment.EnvUtils;
import nexus.io.tio.utils.json.JsonUtils;

public class SjsuProfessorsSearchTest {

  @Test
  public void searchTest() {
    EnvUtils.load();
    new DbConfig().config();
    new SqlTplsConfig().config();
    String question = "Jane Smith";
    EmbeddingRequestVo reoVo = new EmbeddingRequestVo(question, OpenAiModels.text_embedding_3_small);
    EmbeddingResponseVo respVo = OpenAiClient.embeddings(reoVo);
    Float[] embedding = respVo.getData().get(0).getEmbedding();
    String string = Arrays.toString(embedding);

    String sql = SqlTemplates.get("professor.vector_search");

    List<Row> records = Db.find(sql, string, string, string, string, string, string);
    String json = JsonUtils.toJson(records);
    System.out.println(json);
  }
}

String sql = SqlTemplates.get("professor.vector_search"); 返回的 sql 语句如下

SELECT
  id,
  name,
  department,
  job_title,
  email,
  description,
  remark,
  (1-(name_vector <=> ?)) AS name_similarity,
  (1-(description_vector <=> ?)) AS description_similarity,
  (1-(remark_vector <=> ?)) AS remark_similarity
FROM
  rumi_sjsu_professors
ORDER BY
  GREATEST(
    1-(name_vector <=> ?),
    1-(description_vector <=> ?),
    1-(remark_vector <=> ?)
  ) DESC
LIMIT 3;

5. 合并查询参数

在上面的示例中 6 个查询参数都是相同的,能否合并为一个查询参数

SELECT
  id,
  name,
  department,
  job_title,
  email,
  description,
  remark,
  (1-(name_vector <=> ?)) AS name_similarity,
  (1-(description_vector <=> ?)) AS description_similarity,
  (1-(remark_vector <=> ?)) AS remark_similarity
FROM
  rumi_sjsu_professors,
  LATERAL (
    VALUES (
      ?::VECTOR(1536)
    )
  ) AS input(input_vector)
ORDER BY
  GREATEST(
    1-(name_vector <=> ?),
    1-(description_vector <=> ?),
    1-(remark_vector <=> ?)
  ) DESC
LIMIT 3;

查询时只用传入一个参数

List<Row> records = Db.find(sql, string);

SQL 语句解释: 这个 SQL 语句是在进行向量相似度搜索。它从 rumi_sjsu_professors 表中选择数据,并计算输入向量与表中存储的向量之间的相似度。

选择字段包括 id, name, department, job_title, email, description, remark。
计算输入向量与 name_vector, description_vector, remark_vector 的相似度。
使用 LATERAL 子句创建一个临时的输入向量。
按最大相似度降序排序。
限制返回前 3 条结果。

<==> 运算符:

<==> 是 PostgreSQL 中用于计算向量距离的运算符。它返回两个向量之间的余弦距离。值越小表示向量越相似。
1 - 余弦距离,得到的是余弦相似度

6. db 工具类存储向量和查询向量示例

CREATE TABLE rumi_embedding (id bigint PRIMARY KEY,t text, v vector(1536));

import java.util.Arrays;
import java.util.List;
import java.util.Map;
import java.util.stream.Collectors;

import org.junit.Test;
import org.postgresql.util.PGobject;

import nexus.io.data.utils.SnowflakeIdUtils;
import nexus.io.jfinal.plugin.activerecord.Db;
import nexus.io.jfinal.plugin.activerecord.Row;
import nexus.io.jfinal.plugin.utils.PgVectorUtils;
import nexus.io.open.chat.config.DbConfig;
import nexus.io.openai.client.OpenAiClient;
import nexus.io.tio.utils.environment.EnvUtils;
import nexus.io.tio.utils.json.JsonUtils;

public class VectorTest {

  @Test
  public void testSave() {
    EnvUtils.load();
    new DbConfig().config();
    long id = SnowflakeIdUtils.id();

    String name = "Dr. John Doe";
    Float[] embeddingArray = OpenAiClient.embeddingArray(name);
    PGobject pgVector = PgVectorUtils.getPgVector(Arrays.toString(embeddingArray));
    Row row = new Row().set("t", name).set("v", pgVector).set("id", id).setTableName("rumi_embedding");
    Db.save(row);
  }

  @Test
  public void testQuery() {
    EnvUtils.load();
    new DbConfig().config();
    String sql = "select * from rumi_embedding";
    List<Row> records = Db.find(sql);
    List<Map<String, Object>> collect = records.stream().map((e) -> e.toMap()).collect(Collectors.toList());
    String json = JsonUtils.toJson(collect);
    System.out.println(json);
  }
}

output

[{"id":398392108312608768,"t":"Dr. John 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7.直接查询获取向量对象 PGobject

String sql = "select v from rumi_embedding where t=?";
PGobject pGobject = Db.queryFirst(sql, text);
if (pGobject != null) {
  v = pGobject.getValue();
}