On Partial Opimality by Auxiliary Submodular Problems

In this work, we prove several relations between three different energy minimization techniques. A recently proposed methods for determining a provably optimal partial assignment of variables by Ivan Kovtun (IK), the linear programming relaxation approach (LP) and the popular expansion move algorithm by Yuri Boykov. We propose a novel sufficient condition of optimal partial assignment, which is based on LP relaxation and called LP-autarky. We show that methods of Kovtun, which build auxiliary submodular problems, fulfill this sufficient condition. The following link is thus established: LP relaxation cannot be tightened by IK. For non-submodular problems this is a non-trivial result. In the case of two labels, LP relaxation provides optimal partial assignment, known as persistency, which, as we show, dominates IK. Relating IK with expansion move, we show that the set of fixed points of expansion move with any "truncation" rule for the initial problem and the problem restricted by one-vs-all method of IK would coincide -- i.e. expansion move cannot be improved by this method. In the case of two labels, expansion move with a particular truncation rule coincide with one-vs-all method.Comment: 9 pages, 0 figures; Control Systems and Computers #2/2011, Special issue: "Optimal Labeling Problem in Structural Pattern Recognition", pp. 71-78, issn 0130-539

A discriminative view of MRF pre-processing algorithms

While Markov Random Fields (MRFs) are widely used in computer vision, they present a quite challenging inference problem. MRF inference can be accelerated by pre-processing techniques like Dead End Elimination (DEE) or QPBO-based approaches which compute the optimal labeling of a subset of variables. These techniques are guaranteed to never wrongly label a variable but they often leave a large number of variables unlabeled. We address this shortcoming by interpreting pre-processing as a classification problem, which allows us to trade off false positives (i.e., giving a variable an incorrect label) versus false negatives (i.e., failing to label a variable). We describe an efficient discriminative rule that finds optimal solutions for a subset of variables. Our technique provides both per-instance and worst-case guarantees concerning the quality of the solution. Empirical studies were conducted over several benchmark datasets. We obtain a speedup factor of 2 to 12 over expansion moves without preprocessing, and on difficult non-submodular energy functions produce slightly lower energy.Comment: ICCV 201

On Partial Optimality by Auxiliary Submodular Problems

Доказаны определенные соотношения между тремя различными методами минимизации энергии. Предложено новое достаточное условие частичной оптимальности, основанное на LP-релаксации и названное LP-автаркией.Some relations between three different energy minimization techniques are proved. A new sufficient condition of the optimal partial assignment which is based on the LP-relaxation and called LP-autarky is suggested.Доведено певні співвідношення між трьома різними методами оптимізації енергії. Запропоновано нову достатню умову часткової оптимальності, яка базується на LP-релаксації і названа LP-автаркією

Sufficient Condition for Partial Optimality for (max, +)-Labeling Problems and its Usage

Для (max,+)-задач разметки сформулированы достаточные условия оптимальности метки в каждом пикселе изображения. Описан алгоритм, позволяющий определить оптимальные метки в некоторых пикселах и тем самым существенно снизить сложность исходной задачи.Sufficient conditions for the optimal label detection in every pixel are formulate. An algorithm is described which makes it possible to define the optimal labels in some pixels and to decrease essentially the complexity of the original problem.Для (max, +)-задач розмітки сформульовано достатні умови оптимальності мітки у кожному пікселі зображення. Описано алгоритм, що дозволяє визначити оптимальні мітки у деяких пікселах, завдяки чому суттєво зменшується складність вихідної задачі

Maximum Persistency via Iterative Relaxed Inference with Graphical Models

We consider the NP-hard problem of MAP-inference for undirected discrete graphical models. We propose a polynomial time and practically efficient algorithm for finding a part of its optimal solution. Specifically, our algorithm marks some labels of the considered graphical model either as (i) optimal, meaning that they belong to all optimal solutions of the inference problem; (ii) non-optimal if they provably do not belong to any solution. With access to an exact solver of a linear programming relaxation to the MAP-inference problem, our algorithm marks the maximal possible (in a specified sense) number of labels. We also present a version of the algorithm, which has access to a suboptimal dual solver only and still can ensure the (non-)optimality for the marked labels, although the overall number of the marked labels may decrease. We propose an efficient implementation, which runs in time comparable to a single run of a suboptimal dual solver. Our method is well-scalable and shows state-of-the-art results on computational benchmarks from machine learning and computer vision.Comment: Reworked version, submitted to PAM

Optimal Labelling Problems, their Relaxation and Equivalent Transformations

Рассмотрена оптимизационная задача разметок, которая есть обобщением известной задачи о совместимости ограничений, и ее размытая модификация. Описаны два подхода к поиску оптимальной размытой разметки, их достоинства и недостатки. Предложены направления дальнейших исследований.The optimal labeling problem is considered, which is a generalization of the known Constraint Satisfaction Problem, and its relaxed simplification. Two approaches for the relaxed labeling optimization are described as well as their advantages and shortcomings. A direction of future researches is suggested.Розглянуто оптимізаційну задачу розміток, що узагальнює відому задачу про сумісність обмежень, та її розмиту модифікацію. Описано два підходи до пошуку оптимальної розмитої розмітки, їх переваги і недоліки. Наведено напрями подальших досліджень