What is the name of this machine learning problem?

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?