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If we can find a mapping from data to real numbers, ranking the data can be solved by ranking the real numbers. This mapping is called [[utility function]]. For label ranking the mapping is a function <math>f: X \times Y \rightarrow \mathbb{R}\,\!</math> such that <math>y_i \succ_x y_j \Rightarrow f(x,y_i) > f(x,y_j)\,\!</math>. For instance ranking and object ranking, the mapping is a function <math>f: X \rightarrow \mathbb{R}\,\!</math>.
Finding the utility function is a [[Regression analysis|regression]] learning problem{{citation needed|date=March 2025}} which is well developed in machine learning.
===Preference relations===
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==Uses==
Preference learning can be used in ranking search results according to feedback of user preference. Given a query and a set of documents, a learning model is used to find the ranking of documents corresponding to the [[relevance (information retrieval)|relevance]] with this query. More discussions on research in this field can be found in [[Tie-Yan Liu]]'s survey paper.<ref>{{Cite journal |last=Liu |first=Tie-Yan |date=2007 |title=Learning to Rank for Information Retrieval |url=http://www.nowpublishers.com/article/Details/INR-016 |journal=Foundations and Trends in Information Retrieval |language=en |volume=3 |issue=3 |pages=225–331 |doi=10.1561/1500000016 |issn=1554-0669|url-access=subscription }}</ref>
Another application of preference learning is [[recommender systems]].<ref>{{Citation |last1=Gemmis |first1=Marco de |title=Learning Preference Models in Recommender Systems |date=2010 |work=Preference Learning |pages=387–407 |editor-last=Fürnkranz |editor-first=Johannes |url=http://link.springer.com/10.1007/978-3-642-14125-6_18 |access-date=2024-11-05 |place= |publisher=Springer |language=en |doi=10.1007/978-3-642-14125-6_18 |isbn=978-3-642-14124-9 |last2=Iaquinta |first2=Leo |last3=Lops |first3=Pasquale |last4=Musto |first4=Cataldo |last5=Narducci |first5=Fedelucio |last6=Semeraro |first6=Giovanni |editor2-last=Hüllermeier |editor2-first=Eyke|url-access=subscription }}</ref> Online store may analyze customer's purchase record to learn a preference model and then recommend similar products to customers. Internet content providers can make use of user's ratings to provide more user preferred contents.
==References==
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