Thank you for the quick reply. Just to make sure I understand, if X is sparse and n by n with X[0,0] = 1, X_[n-1, n-1]=0 explicitly set (that is only two values are set in X) then this is treated the same for the purposes of the objective function as the all zeros n by n matrix with X[0,0] set to 1? That is all elements of X that are not specified explicitly are assumed to be 0?
It would be really useful if it were possible to have a version of NMF where contributions to the objective function are only counted where the value is explicitly set in X. This is AFAIK the standard formulation for collaborative filtering. Would there be any interest in doing this? In theory it should be a simple modification of the optimisation code. Raphael On Sunday, August 28, 2016, Arthur Mensch <arthur.men...@inria.fr> wrote: > Zeros are considered as zeros in the objective function, not as missing > values - - i.e. no mask in the loss function. > Le 28 août 2016 16:58, "Raphael C" <drr...@gmail.com > <javascript:_e(%7B%7D,'cvml','drr...@gmail.com');>> a écrit : > > What I meant was, how is the objective function defined when X is sparse? > > Raphael > > > On Sunday, August 28, 2016, Raphael C <drr...@gmail.com > <javascript:_e(%7B%7D,'cvml','drr...@gmail.com');>> wrote: > >> Reading the docs for http://scikit-learn.org/st >> able/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 >> >> > _______________________________________________ > scikit-learn mailing list > scikit-learn@python.org > <javascript:_e(%7B%7D,'cvml','scikit-learn@python.org');> > https://mail.python.org/mailman/listinfo/scikit-learn > >
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