TRMF 辅助论文：最小二乘法复现TRMF

2021/11/9 23:41:26

编程Tag： 乘法 plus 复现 20% _% 7D% l% 7Bl% TRMF

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1 目标函数（总）

论文笔记:Temporal Regularized Matrix Factorization forHigh-dimensional Time Series Prediction_UQI-LIUWJ的博客-CSDN博客

$min_{W,X,\Theta} \frac{1}{2}\sum_{(i,t) \in \Omega} (y_it-w_ix_t^T)^2+\lambda_w R_w(W)+\lambda_x R_{AR}(X|\L,\Theta,\eta)+\lambda_\theta R_\theta(\Theta)$

1.1 求解W

我们留下含有W的部分：

$min_{W,X,\Theta} \frac{1}{2}\sum_{(i,t) \in \Omega} (y_it-w_ix_t^T)^2+\lambda_w R_w(W)$

$min_{W_i} \frac{1}{2} [(y_{i1}-w_ix_1^T)^2+\dots+(y_{in}-w_ix_n^T)^2]+ \frac{1}{2}\lambda_w\sum_{i=1}^m w_iw_i^T$

然后对wi求导

线性代数笔记：标量、向量、矩阵求导_UQI-LIUWJ的博客-CSDN博客

$\frac{1}{2} [-2(y_{i1}-w_ix_1^T)x_1+\dots+-2(y_{in}-w_ix_n^T)x_n]+ \frac{1}{2}\lambda_w 2I w_i=0$

$-(y_{i1}x_1+\dots+y_{in}x_n)+ [{(w_ix_1^T)x_1}+\dots+ (w_ix_n^T)x_n]+ \lambda_w I w_i=0$

而 $y_{ij}$ 是一个标量，所以放在xi的左边和右边没有影响

所以

$[{w_ix_1^Tx_1}+\dots+ w_ix_n^Tx_n]+ \lambda_w I w_i=(y_{i1}x_1+\dots+y_{in}x_n)$

也即：

$(\sum_{(i,t)\in \Omega}x_t^Tx_t) w_i+ \lambda_w I w_i=\sum_{(i,t)\in \Omega}y_{it}x_t$

$w_i=[(\sum_{(i,t)\in \Omega}x_t^Tx_t) + \lambda_w I]^{-1}\sum_{(i,t)\in \Omega}y_{it}x_t$

对应的代码如下：（假设sparse_mat表示观测矩阵）

from numpy.linalg import inv as inv
for i in range(dim1):
    #W矩阵的每一行分别计算
    pos0 = np.where(sparse_mat[i, :] != 0)
    #[num_obs] 表示i对应的有示数的数量

    Xt = X[pos0[0], :]
    #[num_obs,rank

    vec0 = sparse_mat[i, pos0[0]] @ Xt
    #sparse_mat[i, pos0[0]] 是一维向量，
    #所以sparse_mat[i, pos0[0]] @ Xt 和 sparse_mat[i, pos0[0]].T @ Xt 是一个意思，
    #输出的都是一个一维向量
    #[rank,1]

    mat0 = inv(Xt.T @ Xt + np.eye(rank))
    #[rank,rank]

    W[i, :] = mat0 @ vec0

其中：

$\sum_{(i,t)\in \Omega}y_{it}x_t$

vec0 = sparse_mat[i, pos0[0]] @ Xt

$[(\sum_{(i,t)\in \Omega}x_i^Tx_i) + \lambda_w I]^{-1}$

mat0 = inv(Xt.T @ Xt + np.eye(rank))

1.2 求解X

我们留下含有X的部分

$min_{W,X,\Theta} \frac{1}{2}\sum_{(i,t) \in \Omega} (y_it-w_ix_t^T)^2+\lambda_w R_w(W)+\lambda_x R_{AR}(X|\L,\Theta,\eta)$

$\lambda_x R_{AR}(X|\L,\Theta,\eta) =\frac{1}{2}\lambda_x \sum_{t=l_d+1}^f[(x_t-\sum_{l \in L}\theta_l \divideontimes x_{t-l})(x_t-\sum_{l \in L}\theta_l \divideontimes x_{t-l})^T]+\frac{1}{2}\lambda_x \eta \sum_{t=1}^f x_t x_t^T$

$\divideontimes$ 表示逐元素乘积 (两个向量a和b，a $\divideontimes$ b可以用diag(a) b表示）

当t=1~ld的时候，我们没有 $R_{AR}$ 什么事情，所以此时我们更新X的方式和之前的W差不多 $min_{W,X,\Theta} \frac{1}{2}\sum_{(i,t) \in \Omega} (y_it-w_ix_t^T)^2+\lambda_x R_x(X)$

同理，X的更新方式为：

$\small x_t=[(\sum_{(i,t)\in \Omega}w_i^Tw_i) + \lambda_x \eta I]^{-1}\sum_{(i,t)\in \Omega}y_{it}w_i$

而当t≥ld+1的时候，我们就需要考虑 $R_{AR}$ 了

对于任意xt（我们令其为xo），他会出现在哪些 $R_{AR}$ 中呢？

首先是 $\frac{1}{2}\lambda_x[(x_o-\sum_{l \in L}\theta_l \divideontimes x_{o-l})(x_o-\sum_{l \in L}\theta_l \divideontimes x_{o-l})^T]$

$=\frac{1}{2}\lambda_x[(x_o-\sum_{l \in L}\theta_l \divideontimes x_{o-l})(x_o^T-(\sum_{l \in L}\theta_l \divideontimes x_{o-l})^T)]$

$\tiny =\frac{1}{2}\lambda_x[x_ox_o^T-(\sum_{l \in L}\theta_l \divideontimes x_{o-l})x_o^T-x_o(\sum_{l \in L}\theta_l \divideontimes x_{o-l})^T+(\sum_{l \in L}\theta_l \divideontimes x_{o-l})(\sum_{l \in L}\theta_l \divideontimes x_{o-l})^T]$

对xo求导，有：

$\small =\frac{1}{2}\lambda_x[2x_o-2\sum_{l \in L}\theta_l \divideontimes x_{o-l}]$

其次，是所有的 $\frac{1}{2}\lambda_x[(x_{o+l}-\sum_{l \in L}\theta_l \divideontimes x_{o})(x_{o+l}-\sum_{l \in L}\theta_l \divideontimes x_{o})^T]$

对每一个l，有用的项就是xo相关的项，于是我们可以写成，对每一个l的

$\frac{1}{2}\lambda_x[(x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l}-\theta_{l} \divideontimes x_{o})(x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l}-\theta_{l} \divideontimes x_{o})^T]$

$\small =\frac{1}{2}\lambda_x[(x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})(x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})^T -\theta_{l} \divideontimes x_{o}(x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})^T -(x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})(\theta_{l} \divideontimes x_{o})^T +\theta_{l} \divideontimes x_{o}(\theta_{l} \divideontimes x_{o})^T]$

对xo求导，有 $\small =\frac{1}{2}\lambda_x [-2\theta_{l} \divideontimes (x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})^T+2(\theta_{l} ^2)\divideontimes x_o]$

于是我们可以写成 $\sum_{l \in L, o+l<T}\frac{1}{2}\lambda_x [-2\theta_{l} \divideontimes (x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})^T+2(\theta_{l} ^2)\divideontimes x_o]$

几部分拼起来，有

$\small \sum_{l \in L, o+l<T, o>l_d}\frac{1}{2}\lambda_x [-2\theta_{l} \divideontimes (x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})^T+2(\theta_{l} \divideontimes \theta_{l} ) \divideontimes x_o]$

$\small +\frac{1}{2}\lambda_x[2x_o-2\sum_{l \in L,o>l_d}\theta_l \divideontimes x_{o-l}]$

$\small +[(\sum_{(i,o)\in \Omega}w_i^Tw_i) + \lambda_x \eta I]x_o-\sum_{(i,o)\in \Omega}y_{io}w_i$

=0

$\small \sum_{l \in L, o+l<T, o>l_d}\lambda_x (\theta_{l} \divideontimes \theta_{l} ) \divideontimes x_o$

$\small +\lambda_x \eta I x_o$

$\small +[(\sum_{(i,o)\in \Omega}w_i^Tw_i) + \lambda_x I]x_o$

=

$\small \sum_{l \in L, o+l<T, o>l_d}\lambda_x [\theta_{l} \divideontimes (x_{o+l}-\sum_{l' \in L-\{l\}}\theta_{l'} \divideontimes x_{o+l'-l})^T]$

$\small +\lambda_x\sum_{l \in L,o>l_d}\theta_l \divideontimes x_{o-l}$

+ $\small \sum_{(i,o)\in \Omega}y_{io}w_i$

所以xo（o≥ld+1）的更新公式为

$\small [(\sum_{(i,o)\in \Omega}w_i^Tw_i) + \lambda_x I+\lambda_x \eta I +\lambda_x\sum_{l \in L, o+l<T, o>l_d} diag(\theta_{l} \divideontimes \theta_{l} )]^{-1}$

3 更新θ

$min_{W,X,\Theta} \frac{1}{2}\sum_{(i,t) \in \Omega} (y_it-w_ix_t^T)^2+\lambda_w R_w(W)+\lambda_x R_{AR}(X|\L,\Theta,\eta)+\lambda_\theta R_\theta(\Theta)$

我们留下和θ (θk)有关的部分

$\small \frac{1}{2}\lambda_x \sum_{t=l_d+1}^f[(x_t-\sum_{l \in L}\theta_l \divideontimes x_{t-l})(x_t-\sum_{l \in L}\theta_l \divideontimes x_{t-l})^T]+\lambda_\theta R_\theta(\Theta)$

$=\frac{1}{2}\lambda_x \sum_{t=l_d+1}^f[(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l}-\theta_k \divideontimes x_{t-k})(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l}-\theta_k \divideontimes x_{t-k})^T]+\lambda_\theta R_\theta(\Theta)$

$=\frac{1}{2}\lambda_x \sum_{t=l_d+1}^f[(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l})(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l})^T$

$-(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l})(\theta_k \divideontimes x_{t-k})^T$

$-(\theta_k \divideontimes x_{t-k})(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l})^T$

$+(\theta_k \divideontimes x_{t-k})(\theta_k \divideontimes x_{t-k})^T]$

$+\lambda_\theta R_\theta(\Theta)$

关于θk求导

$\small \frac{1}{2}\lambda_x \sum_{t=l_d+1}^f[-2(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l})\divideontimes x_{t-k}+2 diag (x_{t-k} \divideontimes x_{t-k})\theta_k]+\lambda_\theta I \theta_k=0$

$\small \theta_k=[\lambda_\theta I + \sum_{t=l_d+1}^f diag (x_{t-k} \divideontimes x_{t-k})\theta_k]^{-1} \sum_{t=l_d+1}^f[(x_t-\sum_{l \in L-\{k\}}\theta_l \divideontimes x_{t-l})\divideontimes x_{t-k}]$

4 总结

$w_i=[(\sum_{(i,t)\in \Omega}x_t^Tx_t) + \lambda_w I]^{-1}\sum_{(i,t)\in \Omega}y_{it}x_t$

x:

t ∈ 1~ld： $\small x_t=[(\sum_{(i,t)\in \Omega}w_i^Tw_i) + \lambda_x \eta I]^{-1}\sum_{(i,t)\in \Omega}y_{it}w_i$

t ≥ld+1 $\small [(\sum_{(i,o)\in \Omega}w_i^Tw_i) + \lambda_x I+\lambda_x \eta I +\lambda_x\sum_{l \in L, o+l<T, o>l_d} diag(\theta_{l} \divideontimes \theta_{l} )]^{-1}$