We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finite-dimensional problems. While for the latter several optimality bounds are known, to our knowledge no such bounds exist in the infinite-dimensional setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula. © 2012 Springer Science+Business Media New York.
|Original language||English (US)|
|Number of pages||19|
|Journal||Journal of Mathematical Imaging and Vision|
|State||Published - Nov 9 2012|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: This publication is partly based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.