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Are there well used examples of feature selection problem where the problem is defined under certain promise? Let's say the task is to select the minimum number of features such that the mutual information of the selected features is under a certain threshold or the mutual information of a certain other type of clustering is over the threshold.

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Yes. The problem of feature selection under constraints is relevant and was studied very well in multiple scenarios. You may want to look at these papers for reference.

  1. Beyond distributive fairness in algorithmic decision making: Feature selection for procedurally fair learning, 2018
  2. Maximum Relevance and Minimum Redundancy Feature Selection Methods for a Marketing Machine Learning Platform, 2019
  3. Fast Feature Selection with Fairness Constraints, 2023
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