TensorPCA

$\langle \mathbf{A}, \mathbf{B} \rangle = \sum_{i_1=1, \ldots, i_n=1}^{i_1=m_1, \ldots, i_n=m_n} \mathbf{A}_{i_1, \ldots, i_n} \mathbf{B}_{i_1, \ldots, i_n},$

$\| \mathbf{A} \| = \sqrt{\langle \mathbf{A}, \mathbf{A} \rangle},$

$\| \mathbf{A} - \mathbf{B} \|.$

$\mathbf{B} = \mathbf{A} \times_k U,$

where $$\mathbf{B}_{i_1, \ldots, i_k, j, i_{k+1}, \ldots, i_n} = \sum_{i=1}^{m_k} A_{i_1, \ldots, i_k, i, i_{k+1}, \ldots, i_n} \times U_{ij}, j = 1, \ldots, m'_k$$

$\left\{\mathbf{X}_{\mathbf{i}} \in \mathbb{R}^{m_1 \times m_2 \times \cdots \times m_n}, i=1,2, \ldots, N\right\}$

$y_i = \mathbf{X}_i \times_1 w^1 \times_2 w^2 \cdots \times_n w^n.$

\begin{aligned} \left(w^1, \ldots, w^n\right)^*= & \underset{f\left(w^1, \ldots, w^n\right)=d}{\arg \min } \sum_{i \neq j} \| \mathbf{X}_{\mathbf{i}} \times_1 w^1 \times_2 w^2 \ldots \times_n w^n -\mathbf{X}_{\mathbf{j}} \times_1 w^1 \times_2 w^2 \ldots \times_n w^n \|^2 W_{i j} \end{aligned}

$f(w^1, \ldots, w^n) = \sum_{i=1}^n \| \mathbf{X}_i \times_1 w^1 \times_2 w^2 \cdots \times w^n \|^2 B_{ii},$

$$B$$ 从惩罚图中产生，即

$B = L^p = D^p - W^p$

\begin{aligned} f\left(w^1, \ldots, w^n\right)= & \sum_{i \neq j} \| \mathbf{X}_{\mathbf{i}} \times_1 w^1 \times_2 w^2 \ldots \times_n w^n-\mathbf{X}_{\mathbf{j}} \times_1 w^1 \times_2 w^2 \ldots \times_n w^n \|^2 W_{i j}^p \end{aligned}