Online NMF - Mathematical Framework

Mathematical Framework

We begin with the initial factorization at timestamp t

V = W*H + E

For simplicity we claim V ~ W*H

We then add the additional data U to matrix V resulting in V'

V' = ( V ) over ( U )

V' = W'*H

V' = ( W1' )*H' over ( W2' )

Also, for previously processed data:

W' = ( W*Λ−1*W1' ) over ( W2' )

H' = W1'−1*Λ*H

Where Λ is a diagonal matrix were Λ_ii equals weight factor h_i

For unprocessed data we need to perform the optimization problem of minimizing J. We have:

J = 1/2 ||V - W*H|| + a*R*H*HT

Where a is a positive integer and R is a symmetry non-negative matrix. This can be done using the following iterative algorithm

w_ij <-- w_ij (V*HT)_ij over (W*H*HT)_ij

h_ij <-- h_ij (WT*V)_ij over (WT*W*H + a*R*H)_ij