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Assuming no particular knowledge, let us consider a system where a mapping $F:\mathbb{R}^n\times \mathbb{R}^k \to \mathbb{R}^{n} \times \{-1,1\}$ describes the transformations of $\mathbb{R}^n$ entities. Additionally, we are provided with a white box model for finding $\mathbf{y}\in \mathbb{R}^n$ such that $F(\mathbf{x},\mathbf{u}) = (\mathbf{y},c)$ for any input $(\mathbf{x},\mathbf{u})$.

Given $M$ training samples, $(\mathbf{x}_m,\mathbf{u}_m,c_m)$ with $\mathbf{x}_m \in \mathbb{R}^n$, $\mathbf{u}_m \in \mathbb{R}^k$, $c_m \in \{\pm 1\}$ such that $F(\mathbf{x}_m,\mathbf{u}_m) = (\mathbf{y}_m,c_m)$ for $m \in \overline{1\ldots M}$, what methods are there for performing the following tasks:

  • find $c^\ast$ such that $F(\mathbf{x}^\ast,\mathbf{u}^\ast) = (\mathbf{y}^\ast,c^\ast)$ for known $\mathbf{x}^\ast,\mathbf{u}^\ast$
  • find $\mathbf{u}^\ast$ such that $F(\mathbf{x}^\ast, \mathbf{u}^\ast) = (\mathbf{y}^\ast, 1)$

For the first task, I suppose that SVMs are one possible option, while for the second one regression could be tried. Are there other classes of solutions that seem more appropriate for this type of problem?

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