Fast Nonnegative Matrix Factorization Algorithms Using Projected Gradient Approaches for Large-Scale Problems

Algorithm 2

(GPSR-BB).

Set  𝐀,𝐗,𝛼_min,𝛼_max,𝜶⁽⁰⁾∈[𝛼_min,𝛼_max]∈ℝ^𝑇%

    Initialization

For  𝑘=1,2,…, % Inner iterations

   Δ^(𝑘)=𝑃_Ω[𝐗^(𝑘)−𝛼^(𝑘)∇_𝑋𝐷_𝐹(𝐘||𝐀𝐗^(𝑘))]−𝐗^(𝑘),

   𝝀^(𝑘)=argmin_𝜆^𝑡(𝑘)_∈[0,1]𝐷_𝐹(𝐘||𝐀(𝐗+Δ^(𝑘)diag{𝝀})),

    where 𝝀=[𝜆_𝑡]∈ℝ^𝑇,

   𝐗^(𝑘+1)=𝐗^(𝑘)+Δ^(𝑘)diag{𝝀},

   𝜸^(𝑘)=diag{(Δ^(𝑘))^𝑇𝐀^𝑇𝐀Δ^(𝑘)},

   If 𝛾^𝑡(𝑘)=0: 𝛼^𝑡(𝑘+1)=𝛼_max,

   Else 𝛼^𝑡(𝑘+1)=min{𝛼_max,max{𝛼_min,

                    [(Δ^(𝑘))^𝑇Δ^(𝑘)]_𝑡𝑡/𝛾^𝑡(𝑘)}},

   End

End  % Inner iterations