Low-Rank Affinity Based Local-Driven Multilabel Propagation

Algorithm 1

Solving problem (2) by ADM.

Inputs: Data matrices {𝑋_𝑖}, parameters 𝜆 and 𝛼.

while not converged do

 (1) Fix the others and update 𝐽₁,…,𝐽_𝐾 by

        𝐽_𝑖=argmin_𝐽𝑖1𝜇𝐽_𝑖∗+12𝐽_𝑖−𝑍_𝑖+𝑊_𝑖𝜇²_𝐹.

 (2) Fix the others and update 𝑆₁,…,𝑆_𝐾 by

        𝑆_𝑖=𝐼+𝑋^𝑇_𝑖𝑋_𝑖⁻¹𝑋^𝑇_𝑖𝑋_𝑖−𝑅_𝑖+𝑍_𝑖+𝑋^𝑇_𝑖𝑌_𝑖+𝑉_𝑖−𝑊_𝑖𝜇.

 (3) Fix the others and update 𝑍 by

      𝑍=argmin_𝑍𝛼‖𝑍‖2𝜇^2,1+12^𝐾_𝑖∑₌₁‖𝑍−𝑀‖²_𝐹,

 where 𝑀 is a 𝐾×𝑁² matrix formed as follows:

         𝑀=(𝐹₁)₁₁(𝐹₁)₁₂⋯(𝐹₁)_𝑛𝑛(𝐹₂)₁₁(𝐹₂)₁₂⋯(𝐹₂)_𝑛𝑛⋮⋮⋱⋮(𝐹_𝐾)₁₁(𝐹_𝐾)₁₂⋯(𝐹_𝐾)_𝑛𝑛,

 where 𝐹_𝑖=(𝐽_𝑖+𝑆_𝑖−(𝑊_𝑖+𝑉_𝑖)𝜇)/2,𝑖=1,…,𝐾.

  (4) Fix the others and update 𝐸₁,…,𝐸_𝐾 by

         𝐸_𝑖=argmin_𝐸𝑖𝜆𝜇𝐸_𝑖2,1+𝐸_𝑖−𝑋_𝑖−𝑋_𝑖𝑆_𝑖+𝑌_𝑖𝜇²_𝐹.

 (5) Update the multipliers

         𝑌_𝑖=𝑌_𝑖+𝜇(𝑋_𝑖−𝑋_𝑖𝑆_𝑖−𝐸_𝑖),

         𝑊_𝑖=𝑊_𝑖+𝜇(𝑍_𝑖−𝐽_𝑖),

         𝑉_𝑖=𝑉_𝑖+𝜇(𝑍_𝑖−𝑆_𝑖).

 (6) Update the parameter 𝜇 by 𝜇=min(𝜌𝜇,10¹⁰)

  (𝜌=1.1 in all experiments).

 (7) Check the convergence condition: 𝑋_𝑖−𝑋_𝑖𝑆_𝑖−𝐸_𝑖→0,𝑍_𝑖−𝐽_𝑖→0 and 𝑍_𝑖−𝑆_𝑖→0, 𝑖=1,…,𝐾.

end while

Output: 𝑍.