[scikit-learn] Does NMF optimise over observed values

[Raphael C]

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.mensch at 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" <drraph at gmail.com
> <javascript:_e(%7B%7D,'cvml','drraph at 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 <drraph at gmail.com
> <javascript:_e(%7B%7D,'cvml','drraph at 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
>>
>>
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