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 Λii equals weight factor hi
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
wij <-- wij (V*HT)ij over (W*H*HT)ij
hij <-- hij (WT*V)ij over (WT*W*H + a*R*H)ij
Read more about this topic: Online NMF
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