Mixture Augmented Lagrange Multiplier Method for Tensor Recovery and Its Applications

Algorithm 1

MALM-TR: MALM for tensor recovery.

Input:  n-mode tensor 𝒜.

Parameters:  𝛼,𝛽,𝛾,𝜆,𝜂,𝜌.

(1) Initialization: 𝑀_(𝑖)=𝐿_(𝑖),𝑁_(𝑖)=0,𝑘=1, 𝜌>0.

(2) Repeat until convergence

(3) for 𝑖=1 to 𝑛

(4) 𝑀^𝑘+1_(𝑖)=𝑈_𝑖𝐷_{𝜆𝑖/𝛼𝑖+𝛾𝑖}(Λ)𝑉^𝑇_𝑖,

 Where 𝑈_𝑖Λ𝑉^𝑇_𝑖=𝛼_𝑖𝐿^𝑘_(𝑖)+𝑌^𝑘_𝑖/𝛼_𝑖+𝛾_𝑖(𝐴^𝑘_(𝑖)−𝑁^𝑘_(𝑖)+𝑊_𝑖^𝑘/𝛾_𝑖)/𝛼_𝑖+𝛾_𝑖.

(5) 𝑁^𝑘+1_(𝑖)=𝒟_{𝜂𝑖/𝛽𝑖+𝛾𝑖}𝛽_𝑖𝑆^𝑘_(𝑖)+𝑍_𝑖^𝑘/𝛽_𝑖+𝛾_𝑖(𝐴^𝑘_(𝑖)−𝑀^𝑘_(𝑖)+𝑊_𝑖^𝑘/𝛾_𝑖)/𝛽_𝑖+𝛾_𝑖.

(6) 𝑌_𝑖^𝑘+1=𝑌^𝑘_𝑖+𝛼_𝑖(𝐿^𝑘+1_(𝑖)−𝑀^𝑘+1_(𝑖)),

(7) 𝑍_𝑖^𝑘+1=𝑍_𝑖^𝑘+𝛽_𝑖(𝑆^𝑘+1_(𝑖)−𝑁^𝑘+1_(𝑖)),

(8) 𝑊_𝑖^𝑘+1=𝑊_𝑖^𝑘+𝛾_𝑖(𝐴_(𝑖)−𝑀^𝑘+1_(𝑖)−𝑁^𝑘+1_(𝑖)).

(9) end for

(10) ℒ^𝑘+1=∑^𝑛_𝑖=1𝛼_𝑖fold𝑀^𝑘+1_(𝑖)−𝑌_𝑖^𝑘+1/𝛼_𝑖,𝑖/∑^𝑛_𝑖=1𝛼_𝑖,

(11) 𝒮^𝑘+1=∑^𝑛_𝑖=1𝛽_𝑖fold𝑁^𝑘+1_(𝑖)−𝑍_𝑖^𝑘+1/𝛽_𝑖,𝑖/∑^𝑛_𝑖=1𝛽_𝑖.

(12) 𝛼=𝜌𝛼,𝛽=𝜌𝛽,𝛾=𝜌𝛾.

(13) 𝑘=𝑘+1.

(14) End

Output:  n-mode tensor ℒ,𝒮.