Sparse IR with user feedback
This is a learning to rank problem, in particular feature selection for learning to rank.

There are many ways to solve the problem. One common method is maximizing feature importance while minimizing similarity. In other words, rank documents based on the most valuable and unique features. This is a variation of Minimum-redundancy-maximum-relevance (mRMR) feature selection. 

To maximize feature importance, first rank all documents using all features, evaluate performance with evaluation measures, and show documents with the highest importance scores.

To minimize similarity, find features that have the least redundancy. Redundancy could be measured as the average value of all mutual information values between each feature pair.

"Feature Selection for Ranking" goes into greater detail. 

Once the features are ordered with mRMR, the user can select specific features. Then re-rank documents based on the user-selected features.