Stable Tensor Principal Component Pursuit: Error Bounds and Efficient Algorithms

. 2019 Dec 3;19(23):5335. doi: 10.3390/s19235335

Algorithm 2 Solving Problem (40) using ADMM.

Input:
The observed tensor $\underline{M}$ , an upper estimation r of $r_{tubal} ({\underline{L}}_{0})$ , the parameters $γ, λ, ρ, δ$ .
1:
Initialize $t = 0$ , ${\underline{L}}^{0} = {\underline{S}}^{0} = {\underline{R}}^{0} = {\underline{Y}}_{1}^{0} = {\underline{Y}}_{2}^{0} = \underline{0} \in R^{n_{1} \times n_{2} \times n_{3}}$ , ${\underline{Q}}^{0} = \underline{0} \in R^{n_{1} \times r \times n_{3}}$ , ${\underline{X}}^{0} = \underline{0} \in R^{r \times n_{2} \times n_{3}}$ .
2:
for $t = 0, \dots, T_{max}$ do
3:
Update $({\underline{L}}^{t + 1}, {\underline{S}}^{t + 1})$ by Equation (42);
4:
Update ${\underline{Q}}^{t + 1}$ by Equation (45);
5:
Update $({\underline{X}}^{t + 1}, {\underline{R}}^{t + 1})$ by Equations (49)–(50);
6:
Update $({\underline{Y}}_{1}^{t + 1}, {\underline{Y}}_{2}^{t + 1})$ by Equation (52);
7:
Check the convergence criteria:
- (i)
  convergence of variables: $∥ {\underline{A}}^{t + 1} - {\underline{A}}^{t} ∥_{\infty} \leq δ, \forall \underline{A} \in {\underline{L}, \underline{S}, \underline{R}, \underline{Q}, \underline{X}}$
- (ii)
  convergence of constraints: $max {∥ {\underline{Q}}^{t + 1} * {\underline{X}}^{t + 1} - {\underline{L}}^{t} ∥_{\infty}, ∥ {\underline{R}}^{t + 1} - {\underline{S}}^{t + 1} ∥_{\infty}} \leq δ$ .
8:
end for
Output:
$(\hat{\underline{L}}, \hat{\underline{S}}) = ({\underline{L}}^{t + 1}, {\underline{S}}^{t + 1})$ .