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I am confronted to a particular type of problem that I do not know how to handle. And I cannot find any literature about this problem.

Problem settings

I have a dataset of $n$ sample points. Each point consists in a set of $m$ vectors of features, named $x^{(i)}_j \in \mathbb R^d$, for $1 \leq i \leq n$ and $1 \leq j \leq m$.

For set of vectors $(x^{(i)}_j)_{1 \leq j \leq m}$, the corresponding values $y^{(i)}_1 , ..., y^{(i)}_m \in \{-1 ; 1\}$ are available. Among the $m$ values, only one is equal to $1$, and the other are equal to $-1$.

What I want the model to do

I would like to learn a model from these data which gives for a new set of vectors $(x^{(0)}_j)_{1 \leq j \leq m}$, the corresponding values $y^{(0)}_1 , ..., y^{(0)}_m$. More precisely, I search for a model which returns: $$ \begin{pmatrix} p(y^{(0)}_1 > 0 | x^{(0)}_1 , ..., x^{(0)}_m ) \\ \vdots \\ p(y^{(0)}_m > 0 | x^{(0)}_1 , ..., x^{(0)}_m ) \end{pmatrix}. $$

My questions

Does this problem have a name in the statistical literature?

Could you point me papers which treat that issue or algorithms which are able to learn this particular classification problem?

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You should consider multinomial logistic regression. It is, I think, exactly what you are looking for, assuming that all d choices are available to all subjects.

If that assumption doesn't hold, you should look in the literature for discrete choice models. Be wary of the assumptions of these models, many require behavioral assumptions that are dubious.

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  • $\begingroup$ The products can different for each user i, so that I think it cannot be seen as a multilabel classification problem. $\endgroup$ – Pop Nov 23 '15 at 9:44
  • $\begingroup$ Ah, in that case you should look up the literature on Discrete Choice models. They have a variety of (conditional) multinomial logistic models that work when the set of available items differs. Just be careful when choosing a model, that it makes sense for your situation (I don't trust many of the utility theory models, but that might be what you are stuck with). Here is a nice set of examples of discrete choice models in R. $\endgroup$ – AlaskaRon Nov 23 '15 at 9:56

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