Why -- in supervised learning literature -- do we assume the sample space to be the cartesian product of input space i.e., $\mathcal{Z}=\mathcal{X}\times\mathcal{Y}$. Since there is a one-to one mapping in most cases. Why can't we consider $\mathcal{Z}=\{(x,y=g(x)): x\in\mathcal{X}\}$ where $g$ is some data generating function. It reduces the sample space size considerably. What I want to say is that is there a specific reason why we make the cartesian product supposition?
2026-03-28 14:44:23.1774709063
why do we assume the sample space to be the cartesian product of input and output space?
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