Suppose $\varphi: X \rightarrow Y$ is a function between feature spaces that I want to model. I want to produce two easily computable functions $\phi_X: X \rightarrow \mathbb{R}^m$ and $\phi_Y: Y \rightarrow \mathbb{R}^n$ so the distance $|\phi_X(x) - \phi_X(x')|$ for $x, x'$ in $X$ is correlated as well as possible with $|\phi_Y(\varphi(x)) - \phi_Y(\varphi(x'))|$.

My question is simple: is there any research going on about questions like this? The problem seems awkwardly stuck between a clustering problem and a classification problem. I don't need an ironclad way of approaching the problem (there almost certainly isn't one), but I was wondering if the problem had any better classification than as a kinda-regression.

EDIT: As per D.W.'s comment, my condition is poorly stated as it stands. A condition that better expresses what I meant is to find a method that, given a point $x \in X$, a finite set of points $y_i \in Y$, and an assurance that $\varphi(x) = y_i$ for some $i$, to determine the most likely $y_i$ that fits this bill. But this is straight machine-ranking, solving the question.

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    $\begingroup$ 1. I think you'll need to define your problem more carefully. For instance, $\phi_X(x)=0$ for all $x$ and $\phi_Y(y)=0$ for all $y$ provides an excellent correlation. I suspect you must have some additional requirements or constraints that you're not telling us about. 2. What kind of a space are $X,Y$? $\endgroup$ – D.W. Sep 22 '15 at 22:27
  • $\begingroup$ Your first comment is a good point, and I reformulated the condition to be a bit more coherent. However, having done so, I realized that the answer to my question was simply machine ranking. Thanks for the push in the right direction! $\endgroup$ – Alexander Smith Sep 23 '15 at 2:26

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