Nonnegative Matrix Factorizations Performing Object Detection and Localization

Algorithm 3

Local nonnegative matrix factorization update rules.

Initialize nonnegative matrices 𝑊⁽⁰⁾ and 𝐻⁽⁰⁾

While Stopping criteria are not satisfied do

 𝐻←(𝐻⊙(𝑊^⊤(𝑉⊘𝑊𝐻)))^⋅(1/2)

 𝑊←𝑊⊙((𝑉⊘𝑊𝐻)𝐻^⊤)

 𝑊←𝑊diag(‖𝑊_∗1‖₁,‖𝑊_∗2‖₁,…,‖𝑊_∗𝑟‖₁)⁻¹

end while

{⊙ and ⊘ denotes the Hadamard product and the element-wise division,

respectively, (⋅)^⋅(1/2) denotes the element-wise square root operation,

diag(‖𝑊_∗1‖₁,‖𝑊_∗2‖₁,…,‖𝑊_∗𝑟‖₁) indicates the 𝑟×𝑟 diagonal matrix

whose diagonal elements are the 1-norm of the column basis vectors in

𝑊}