Probabilistic graphical models provide a powerful framework for representing image structures. Based on this concept, many inference and learning algorithms have been
developed. However, both algorithm classes are NP-hard combinatorial problems in the general case. As a consequence, relaxation methods were developed to
approximate the original problems but with the benefit of being computationally efficient. In this work we consider the learning problem on binary graphical models and their relaxations. Two novel methods for determining the model parameters in discrete energy functions from training data are proposed. Learning the model parameters overcomes the problem of heuristically determining them.
Motivated by common learning methods which aim at minimizing the training error measured by a loss function we develop a new learning method similar in fashion to structured SVM. However, computationally more efficient. We term this method
as linearized approach (LA) as it is restricted to linearly dependent potentials. The linearity of LA is crucial to come up with a tight convex relaxation, which allows
to use off-the-shelf inference solvers to approach subproblems which emerge from solving the overall problem.
However, this type of learning methods almost never yield optimal solutions or perfect performance on the training data set. So what happens if the learned graphical
model on the training data would lead to exact ground segmentation? Will this give a benefit when predicting?
Motivated by the idea of inverse optimization, we take advantage of inverse linear programming to develop a learning approach, referred to as inverse linear
programming approach (invLPA). It further refines the graphical models trained, using the previously introduced methods and is capable to perfectly predict ground
truth on training data. The empirical results from implementing invLPA give answers to our questions posed before. LA is able to learn both unary and pairwise potentials jointly while with invLPA
this is not possible due to the representation we use. On the other hand, invLPA does not rely on a certain form for the potentials and thus is flexible in the choice of
the fitting method. Although the corrected potentials with invLPA always result in ground truth
segmentation of the training data, invLPA is able to find corrections on the foreground segments only. Due to the relaxed problem formulation this does not affect the final
segmentation result. Moreover, as long as we initialize invLPA with model parameters of a learning method performing sufficiently well, this drawback of invLPA does not
significantly affect the final prediction result.
The performance of the proposed learning methods is evaluated on both synthetic and real world datasets. We demonstrate that LA is competitive in comparison to other parameter learning methods using loss functions based on Maximum a Posteriori Marginal (MPM) and Maximum Likelihood Estimation (MLE). Moreover, we illustrate the benefits of learning with inverse linear programming. In a further experiment we demonstrate the versatility of our learning methods by applying LA to learning motion segmentation in video sequences and comparing it to state-of-the-art segmentation algorithms
