Computational Intelligence and Neuroscience

Research Article

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

Algorithm 3

(NMF-PSESOP).
Set   𝐀 , 𝐱 𝑡 ( 0 ) , 𝑝 % Initialization
For   𝑘 = 1 , 2 , … , % Inner iterations
    𝐝 1 ( 𝑘 ) = 𝐱 ( 𝑘 ) − 𝐱 ( 0 ) ,
    𝐠 ( 𝑘 ) = ∇ 𝐱 𝑡 𝐷 𝐹 ( 𝐲 𝑡 | | 𝐀 𝐱 𝑡 ) ,
    𝐆 ( 𝑝 ) = [ 𝐠 ( 𝑘 − 1 ) , 𝐠 ( 𝑘 − 2 ) , … , 𝐠 ( 𝑘 − 𝑝 ) ] ∈ ℝ 𝐽 × 𝑝 ,
    𝑤 𝑘 = ⎧ ⎪ ⎨ ⎪ ⎩ 1 i f 𝑘 = 1 , 1 / 2 + 1 / 4 + 𝑤 2 𝑘 − 1 i f 𝑘 > 1 ,
    𝐰 ( 𝑘 ) = [ 𝑤 𝑘 , 𝑤 𝑘 − 1 , … , 𝑤 𝑘 − 𝑝 + 1 ] 𝑇 ∈ ℝ 𝑝 ,
    𝐝 2 ( 𝑘 ) = 𝐆 ( 𝑝 ) 𝐰 ( 𝑘 ) ,
    𝐃 ( 𝑘 ) = [ 𝐝 1 ( 𝑘 ) , 𝐝 2 ( 𝑘 ) , 𝐠 ( 𝑘 ) , 𝐆 ( 𝑝 ) ] ,
    𝜶 ∗ ( 𝑘 ) = a r g m i n 𝛼 𝐷 𝐹 ( 𝐲 𝑡 | | 𝐀 ( 𝐱 𝑡 ( 𝑘 ) + 𝐃 ( 𝑘 ) 𝜶 ( 𝑘 ) ) ) ,
    𝐱 ( 𝑘 + 1 ) = 𝑃 Ω [ 𝐱 ( 𝑘 ) + 𝐃 ( 𝑘 ) 𝜶 ∗ ( 𝑘 ) ]
End  % Inner iterations