Traffic Volume Data Outlier Recovery via Tensor Model

Algorithm 1

ADMM-TR: ADMM for tensor recovery.

Input: n-mode tensor 𝒜

Parameters: 𝛼,𝛽,𝛾,𝜆,𝜂,𝜌

(1) Initialization: ℒ⁽⁰⁾=𝒜,  𝒮⁽⁰⁾=0,  𝑀_𝑖=ℒ_(𝑖),  𝑁_𝑖=0,  𝑘=1

(2) Repeat until convergence

(3) for 𝑖=1 to 𝑛

(4)  𝑀_𝑖^(𝑘+1)=𝑈_𝑖𝐷_{𝜆𝑖/𝛼𝑖}(Λ)𝑉^𝑇_𝑖

       where 𝑈_𝑖Λ𝑉^𝑇_𝑖=𝑃_𝑖ℒ^(𝑘)+𝑌_𝑖^(𝑘)/𝛼_𝑖

(5)  𝑁_𝑖^(𝑘+1)=𝐷_{𝜂𝑖/𝛽𝑖}(𝑃_𝑖𝒮^(𝑘)+𝑍_𝑖^(𝑘)/𝛽_𝑖)

(6)  𝑌_𝑖^(𝑘+1)=𝑌_𝑖^(𝑘)+𝛼_𝑖(𝑃_𝑖ℒ^(𝑘)−𝑀_𝑖^(𝑘+1))

(7)  𝑍_𝑖^(𝑘+1)=𝑍_𝑖^(𝑘)+𝛽_𝑖(𝑃_𝑖𝒮^(𝑘)−𝑁_𝑖^(𝑘+1))

(8) end for

(9) ℒ^(𝑘+1)=𝒜−𝒮^(𝑘)−𝛾∑^𝑛_𝑖=1𝑃^𝑇_𝑖𝑌_𝑖^(𝑘+1)−𝛼_𝑖𝑀_𝑖^(𝑘+1)1+𝛾∑^𝑛_𝑖=1𝛼_𝑖

(10) 𝒮^(𝑘+1)=𝒜−ℒ^(𝑘+1)−𝛾∑^𝑛_𝑖=1𝑃^𝑇_𝑖𝑍_𝑖^(𝑘+1)−𝛼_𝑖𝑁_𝑖^(𝑘+1)1+𝛾∑^𝑛_𝑖=1𝛽_𝑖

(11) 𝛼=𝜌𝛼,  𝛽=𝜌𝛽

(12) End

(13)  𝑘=𝑘+1

Output: n-mode tensor ℒ,𝒮