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#### Estimating Maximally Probable Constrained Relations by Mathematical Programming

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##### Fulltext (public)

1408.0838.pdf

(Preprint), 522KB

##### Supplementary Material (public)

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##### Citation

Qu, L., & Andres, B. (2014). Estimating Maximally Probable Constrained Relations by Mathematical Programming. Retrieved from http://arxiv.org/abs/1408.0838.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0024-D324-6

##### 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.