@online{qu-2014,
TITLE = {Estimating Maximally Probable Constrained Relations by Mathematical Programming},
AUTHOR = {Qu, Lizhen and Andres, Bj{\"o}rn},
LANGUAGE = {eng},
URL = {http://arxiv.org/abs/1408.0838},
EPRINT = {1408.0838},
EPRINTTYPE = {arXiv},
YEAR = {2014},
ABSTRACT = {Estimating a constrained relation is a fundamental problem in machine learning. Special cases are classification (the problem of estimating a map from a set of to-be-classified elements to a set of labels), clustering (the problem of estimating an equivalence relation on a set) and ranking (the problem of estimating a linear order on a set). We contribute a family of probability measures on the set of all relations between two finite, non-empty sets, which offers a joint abstraction of multi-label classification, correlation clustering and ranking by linear ordering. Estimating (learning) a maximally probable measure, given (a training set of) related and unrelated pairs, is a convex optimization problem. Estimating (inferring) a maximally probable relation, given a measure, is a 01-linear program. It is solved in linear time for maps. It is NP-hard for equivalence relations and linear orders. Practical solutions for all three cases are shown in experiments with real data. Finally, estimating a maximally probable measure and relation jointly is posed as a mixed-integer nonlinear program. This formulation suggests a mathematical programming approach to semi-supervised learning.},
}