
DeAnti
Same on the other side
$$ P(u,i) = p _u^Tq _i = \sum _{f=1}^{F}p _{uf}q _{if} $$
$p _{u}$
与 $q _{i}$
是向量, $p _u$
的转置 $p _u^T$
与 $q _i$
相乘为一值$$ loss = \sum _{(u,i) \in D} \big(p(u,i)-p^{LFM}(u,i)\big)^2 $$
$p^{LFM}(u,i)$
是LFM算法预估值$$
\begin{equation}
\widetilde{J}(w;X,y) = J(w;X,y) + \alpha\Omega(w) \\[20px]
\left\{
\begin{aligned}
\Omega(w) &= \|w\| _1 = \sum _i|w _i| & (l1正则化)\\[8px]
\Omega(w) &= \|w\| _2 = \sum _iw _i^2 & (l2正则化)
\end{aligned}
\right .
\end{equation}
$$
$$ loss = \sum _{(u,i) \in D}\big(p(u,i) - \sum _{f=1}^Fp _{uf}q _{if}\big)^2 + \partial|p _u|^2 + \partial|q _i|^2 $$
$$ \frac{\partial loss}{\partial p _{uf}} = -2\big(p(u,i) - p^{LFM}(u,i)\big)q _{if} + 2\partial p _{uf} $$
$$ \frac{\partial loss}{\partial q _{if}} = -2\big(p(u,i) - p^{LFM}(u,i)\big)p _{uf} + 2\partial q _{if} $$
$$
\begin{equation}
\left\{
\begin{aligned}
p _{uf} &= p _{uf} - \beta\frac{\partial loss}{\partial p _{uf}} \\
q _{if} &= q _{if} - \beta\frac{\partial loss}{\partial q _{if}}
\end{aligned}
\right .
\end{equation}
$$
复杂度 | LFM | CF |
---|---|---|
时间 | O(dnF) | O(mk^2) |
空间 | O(n) | O(n^2) |
离线计算 | 响应及时 |