Pattern Expression Nonnegative Matrix Factorization: Algorithm and Applications to Blind Source Separation

Algorithm 1

Learning algorithm.

Algorithm parameters: 𝛼,𝛽;

Input: an n by m nonnegative observation matrix V;

Output: an n by r nonnegative matrix W and an r by m

nonnegative matrix H.

Step 1: set 𝑡=0, and generate nonnegative matrix 𝑊⁰ and

𝐻⁰ at random;

Step 2: Update 𝐻 from 𝐻^𝑡 to 𝐻^𝑡+1 by

𝐻^𝑡+1=𝐻^𝑡⊗(𝑊^𝑡)^𝑇𝑉(𝑊^𝑡)^𝑇(𝑊^𝑡)(𝐻^𝑡,)+𝛽𝐼

𝑊^𝑡+1=𝑊^𝑡⊗𝑉(𝐻^𝑡)^𝑇𝑊^𝑡(𝐻^𝑡)(𝐻^𝑡)^𝑇+𝛼𝑊^𝑡𝑀,

where I is an r by m matrix full of elements being 1s, and M

is an r by r matrix with all elements being 1s except diagonal

elements being zeros.

Step 3: Increment t by 𝑡=𝑡+1 and go to step 2 until 𝐻^𝑡+1

and 𝑊^𝑡+1 converge.