Randomized SVD Methods in Hyperspectral Imaging

Algorithm 1

Construct an orthonormal matrix 𝑄 that approximates the range of matrix 𝐴Ω.

Input: An 𝑚×𝑛 matrix 𝐴 and 𝑎 precision measure 𝜖.

Output: An 𝑚×𝑘 matrix 𝑄 and rank 𝑘.

Initialize 𝑄 as an empty matrix, 𝑒=1 and 𝑘=0.

while 𝑒>𝜖 do

 (1)  𝑘=𝑘+1.

 (2)  𝑦_𝑖=𝐴𝜔_𝑖, where 𝜔_𝑖 is a Gaussian random vector.

 (3)  𝑞_𝑖=(𝐼−𝑄𝑄^𝑇)𝑦_𝑖.

 (4)  𝑞_𝑖=𝑞_𝑖/‖𝑞_𝑖‖.

 (5)  𝑄←[𝑄𝑞_𝑖].

 (6)  Ω←[Ω𝜔_𝑖].

 (7)  Compute error 𝑒=‖𝐴−𝑄𝑄^𝑇𝐴‖_𝐹/‖𝐴‖_𝐹.

end