FM和FFM学习

FM后半部分公式化简，时间复杂度从

O (k n^{2}) 降 到 O (k n)

$O(kn^2)降到O(kn)$

\begin{aligned} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (V_{i}^{T} V_{j}) x_{i} x_{j} \end{aligned}

$\begin{aligned} \sum_{i=1}^{n-1}\sum_{j=i+1}^n(V_i^TV_j)x_ix_j \end{aligned}$

\begin{aligned} = \frac{1}{2} (\sum_{i = 1}^{n} \sum_{j = 1}^{n} (V_{i}^{T} V_{j}) x_{i} x_{j} - \sum_{i = 1}^{n} (V_{i}^{T} V_{i}) x_{i} x_{i}) \end{aligned}

$\begin{aligned} & = \frac{1}{2}\left(\sum_{i=1}^n\sum_{j=1}^n(V_i^TV_j)x_ix_j-\sum_{i=1}^n(V_i^TV_i)x_ix_i\right)\\\end{aligned}$

\begin{aligned} = \frac{1}{2} (\sum_{i = 1}^{n} \sum_{j = 1}^{n} \sum_{l = 1}^{k} v_{i l} v_{j l} x_{i} x_{j} - \sum_{i = 1}^{n} \sum_{l = 1}^{k} v_{i l}^{2} x_{i}^{2}) \end{aligned}

$\begin{aligned} & =\frac{1}{2}\left(\sum_{i=1}^n\sum_{j=1}^n\sum_{l=1}^kv_{il}v_{jl}x_ix_j-\sum_{i=1}^n\sum_{l=1}^k v_{il}^2x_i^2\right)\\\\\end{aligned}$

\begin{aligned} = \frac{1}{2} \sum_{l = 1}^{k} (\sum_{i = 1}^{n} (v_{i l} x_{i}) \sum_{j = 1}^{n} (v_{j l} x_{j}) - \sum_{i = 1}^{n} v_{i l}^{2} x_{i}^{2}) \end{aligned}

$\begin{aligned} & =\frac{1}{2}\sum_{l=1}^k\left(\sum_{i=1}^n(v_{il}x_i)\sum_{j=1}^n(v_{jl}x_j)-\sum_{i=1}^nv_{il}^2x_i^2\right)\\\end{aligned}$

\begin{aligned} = \frac{1}{2} \sum_{l = 1}^{k} ({(\sum_{i = 1}^{n} (v_{i l} x_{i}))}^{2} - \sum_{i = 1}^{n} v_{i l}^{2} x_{i}^{2}) \end{aligned}

$\begin{aligned} &=\frac{1}{2}\sum_{l=1}^k\left(\left(\sum_{i=1}^n(v_{il}x_i)\right)^2-\sum_{i=1}^nv_{il}^2x_i^2\right)\\ \end{aligned}$
第一行：对称矩阵，减去对角线元素，除以2。得到最终值。
第二行：把大V展开变成小v

点击率预估算法：FM与FFM
深入FFM原理与实践
论文：Factorization Machines

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