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Given

  1. A Guassian process $w(\mu^*,\Sigma^*)$
  2. $n$ observations of the output
  3. And $n$ potential inputs.

But the assignment of the inputs to the observations is unknown. The goal is to find a inputs-to-outputs matching that leads to the highest probability.

Formally, the objective is to find a permutation of the given input values $X'_{i}\in Per\{x'_{1},...,x'_{n}\}$ (where $Per$ is the permutation group includes all possible permutations) that maximize the probability of observing the given vector of output values $Y'=[y'_{1}...y'_{n}]$ at the corresponding input. Or:

$ argmax_{i}\{Pr(Y'|X'_{i},w)\}~~/or/~~argmax_{i}\{\mathcal N(\mu^*_{X'_{i}},\Sigma^*_{X'_{i}})_{Y'}\}$

Is there any solution for the above problem better than brute-force?

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