IVF+PQ原理概述

1️⃣$k\text{-Means}$分簇方法

1. 含义：一种无监督学习，用于将数据集分为$k$个簇(每簇一个质心)，使同簇点靠近/异簇点远离

2. 流程：

   

2️⃣$\text{PQ}$算法流程

0. 给定$k$个$D$维向量

   $\{\begin{array}{l}{\text{v}}_1=[x_{11},x_{12},x_{13},x_{14},...,x_{1D}]\\ \\ {\text{v}}_2=[x_{21},x_{22},x_{23},x_{24},...,x_{2D}]\\ \\ .........\\ \\ {\text{v}}_k=[x_{k1},x_{k2},x_{k3},x_{k4},...,x_{aD}]\end{array}\leftrightarrow \{\begin{array}{l}{\text{v}}_1=\{[x_{11},x_{12},x_{13}],[x_{14},x_{15},x_{16}],...,[x_{1(D-1)},x_{1(D-1)},x_{1D}]\}\\ \\ {\text{v}}_2=\{[x_{21},x_{22},x_{23}],[x_{24},x_{25},x_{26}],...,[x_{2(D-1)},x_{2(D-1)},x_{2D}]\}\\ \\ .........\\ \\ {\text{v}}_k=\{[x_{k1},x_{k2},x_{k3}],[x_{k4},x_{k5},x_{k6}],...,[x_{k(D-1)},x_{k(D-1)},x_{kD}]\}\end{array}$

1. 分割子空间：将$D$维向量分为$M$个$\frac{D}{M}$维向量

   $子空间1\{\begin{array}{l}{\text{v}}_{11}=[x_{11},x_{12},x_{13}]\\ \\ {\text{v}}_{21}=[x_{21},x_{22},x_{23}]\\ \\ .........\\ \\ {\text{v}}_{k1}=[x_{k1},x_{k2},x_{k3}]\end{array}\&子空间2\{\begin{array}{l}{\text{v}}_{12}=[x_{14},x_{15},x_{16}]\\ \\ {\text{v}}_{22}=[x_{24},x_{25},x_{26}]\\ \\ .........\\ \\ {\text{v}}_{k2}=[x_{k4},x_{k5},x_{k6}]\end{array}\&...\&子空间M\{\begin{array}{l}{\text{v}}_{1M}=[x_{1(D-1)},x_{1(D-1)},x_{1D}]\\ \\ {\text{v}}_{2M}=[x_{2(D-1)},x_{2(D-1)},x_{2D}]\\ \\ .........\\ \\ {\text{v}}_{kM}=[x_{k(D-1)},x_{k(D-1)},x_{kD}]\end{array}$

2. 生成$\text{PQ}$编码:

   $子空间1\{\begin{array}{l}{\text{v}}_{11}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{11}\\ \\ {\text{v}}_{21}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{21}\\ \\ .........\\ \\ {\text{v}}_{k1}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{k1}\end{array}\&子空间2\{\begin{array}{l}{\text{v}}_{12}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{12}\\ \\ {\text{v}}_{22}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{22}\\ \\ .........\\ \\ {\text{v}}_{k2}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{k2}\end{array}\&...\&子空间M\{\begin{array}{l}{\text{v}}_{1M}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{1M}\\ \\ {\text{v}}_{2M}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{2M}\\ \\ .........\\ \\ {\text{v}}_{kM}\stackrel{替代}{\leftarrow }{\text{Centriod}}_{kM}\end{array}$

   • 聚类：在每个子空间上运行$k\text{-Means}$算法(一般$k\text{=}256$)$\to$每个${\text{v}}_{ij}$都会分到一个$\frac{D}{M}$维的质心

   • 编码：将每个子向量${\text{v}}_{ij}$所属质心的索引作为其$\text{PQ}$编码，并替代原有子向量

3. 生成最终的压缩向量→{v1~={Centriod11,Centriod12,...,Centriod1M}v2~={Centriod21,Centriod22,...,Centriod2M}            .........vk~={Centriodk1,Centriodk2,...,CentriodkM}\small\to\begin{cases} \widetilde{\textbf{v}_{1}}=\{\text{Centriod}_{11},\text{Centriod}_{12},...,\text{Centriod}_{1M}\}\\\\ \widetilde{\textbf{v}_{2}}=\{\text{Centriod}_{21},\text{Centriod}_{22},...,\text{Centriod}_{2M}\}\\\\ \,\,\,\,\,\,\,\,\,\,\,\,.........\\\\ \widetilde{\textbf{v}_{k}}=\{\text{Centriod}_{k1},\text{Centriod}_{k2},...,\text{Centriod}_{kM}\} \end{cases}→⎩⎨⎧​v1​​={Centriod11​,Centriod12​,...,Centriod1M​}v2​​={Centriod21​,Centriod22​,...,Centriod2M​}.........vk​​={Centriodk1​,Centriodk2​,...,CentriodkM​}​

   • 存储阶段：存储的内容实质上是质心索引，每个向量只占用$M$维
   • 使用阶段：所有的质心索引被解压为质心，每个向量维度又恢复$M\text{×}\frac{D}{M}\text{=}D$维

3️⃣$\text{IVF+PQ}$原理

1. 离线索引阶段：
   • 构建$\text{IVF}$：使用$\text{K-Menas}$将原始向量集合划分为$n$簇(即$n$个质心)
   • 簇内压缩：对每个簇执行$\text{PQ}$压缩，即将每个簇内向量替换为质心索引
2. 在线查询阶段：
   • $\text{IVF}$部分：计算与查询$q$与所有簇质心的距离，由此选定前$n_{\text{probe}}$个簇的所有向量
   • $\text{PQ}$部分：由质心索引还原选定向量$\text{→}$计算$q$之的距离(遍历子空间)$\text{dist}(q,v)\text{≈}\sum _{i=1}^M\text{dist}\left(q_i,c_{ji}\right)\text{→}$返回最近邻