Algorithm 1. Dynamic Traffic Flow Prediction

Input: sparse traffic flow tensor X with data removed in the prediction horizon.

Output: traffic prediction tensor $\tilde{\mathbf{X}}$.

1. Set the order of the core tensor $\mathsf{\Phi }$.

2. Initialize the core tensor $\mathsf{\Phi }$.

3. Initialize each factor matrix $A^{(1)},A^{(2)},\cdots ,A^{(N)};$

4. Utilize the factor matrices to calculate the core tensor $\mathsf{\Phi }$, according to the formula in (6).

5. Pre-calculate the tensor $\mathbf{Y}=\mathbf{W}\ast \mathbf{X}$ with norm $\gamma =\Vert \mathbf{Y}\Vert$.

6. Calculate tensor $\mathbf{Z}=\mathbf{W}\ast ⟦\mathsf{\Phi };{\mathbf{A}}^{(1)},\cdots ,{\mathbf{A}}^{(N)}⟧$.

7. Calculate the tensor $\mathbf{T}=\mathbf{Y}-\mathbf{Z}$.

8. Calculate $f=\frac{1}{2}\gamma -<\mathbf{Y},\mathbf{Z}>+\frac{1}{2}{\Vert \mathbf{Z}\Vert }^2$.

9. Calculate the gradient matrix as $G^{(n)}=-{\mathbf{T}}_{(n)}{(\mathsf{\Phi }{\times }_1{A}^{(1)}\times \cdots {\times }_{n-1}{A}^{(n-1)}{\times }_{n+1}{A}^{(n+1)}\times \cdots {\times }_N{A}^{(N)})}_{(n)}^{\mathbf{T}}$

10. Utilizing the gradient decent algorithm to get the core tensor $\mathsf{\Phi }$ and the factor matrices ${\mathbf{A}}^{(1)},\cdots ,{\mathbf{A}}^{(N)}$.

11. Calculate the filled tensor $\tilde{\mathbf{X}}=\mathsf{\Phi }{\times }_1{A}^{(1)}{\times }_2{A}^{(2)}\times \cdots {\times }_N{A}^{(N)}$

12. Return $\tilde{\mathbf{X}}$