What is the name of this machine learning problem?

Question

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?

Answer 1

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.