[scikit-learn] Does NMF optimise over observed values

[Raphael C]

Reading the docs for
http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.NMF.html
it
says

The objective function is:

0.5 * ||X - WH||_Fro^2
+ alpha * l1_ratio * ||vec(W)||_1
+ alpha * l1_ratio * ||vec(H)||_1
+ 0.5 * alpha * (1 - l1_ratio) * ||W||_Fro^2
+ 0.5 * alpha * (1 - l1_ratio) * ||H||_Fro^2

Where:

||A||_Fro^2 = \sum_{i,j} A_{ij}^2 (Frobenius norm)
||vec(A)||_1 = \sum_{i,j} abs(A_{ij}) (Elementwise L1 norm)

This seems to suggest that it is optimising over all values in X even
if X is sparse.   When using NMF for collaborative filtering we need
the objective function to be defined over only the defined elements of
X. The remaining elements should effectively be regarded as missing.


What is the true objective function NMF is using?


Raphael
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