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Generalized roof duality and bisubmodular functions
, 2010
"... Consider a convex relaxation ˆ f of a pseudo-boolean function f. We say that the relaxation is totally half-integral if ˆ f(x) is a polyhedral function with halfintegral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form xi = xj, xi = 1 ..."
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Consider a convex relaxation ˆ f of a pseudo-boolean function f. We say that the relaxation is totally half-integral if ˆ f(x) is a polyhedral function with halfintegral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form xi = xj, xi = 1 − xj, and xi = γ where γ ∈ {0, 1, 1 2} is a constant. A well-known example is the roof duality relaxation for quadratic pseudo-boolean functions f. We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations ˆ f by establishing a one-to-one correspondence with bisubmodular functions. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality.
Reflection methods for user-friendly submodular optimization
"... Recently, it has become evident that submodularity naturally captures widely oc-curring concepts in machine learning, signal processing and computer vision. Con-sequently, there is need for efficient optimization procedures for submodular func-tions, especially for minimization problems. While gener ..."
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Recently, it has become evident that submodularity naturally captures widely oc-curring concepts in machine learning, signal processing and computer vision. Con-sequently, there is need for efficient optimization procedures for submodular func-tions, especially for minimization problems. While general submodular minimiza-tion is challenging, we propose a new method that exploits existing decomposabil-ity of submodular functions. In contrast to previous approaches, our method is neither approximate, nor impractical, nor does it need any cumbersome parame-ter tuning. Moreover, it is easy to implement and parallelize. A key component of our method is a formulation of the discrete submodular minimization problem as a continuous best approximation problem that is solved through a sequence of reflections, and its solution can be easily thresholded to obtain an optimal discrete solution. This method solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruc-tion. In our experiments, we illustrate the benefits of our method on two image segmentation tasks. 1
A Primal-Dual Algorithm for Higher-Order Multilabel Markov Random Fields
"... Graph cuts method such as α-expansion [4] and fu-sion moves [22] have been successful at solving many optimization problems in computer vision. Higher-order Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, es-pecially for multilabel MRF’s ..."
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Cited by 3 (0 self)
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Graph cuts method such as α-expansion [4] and fu-sion moves [22] have been successful at solving many optimization problems in computer vision. Higher-order Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, es-pecially for multilabel MRF’s (i.e. more than 2 labels). In this paper we propose a new primal-dual energy minimiza-tion method for arbitrary higher-order multilabel MRF’s. Primal-dual methods provide guaranteed approximation bounds, and can exploit information in the dual variables to improve their efficiency. Our algorithm generalizes the PD3 [19] technique for first-order MRFs, and relies on a variant of max-flow that can exactly optimize certain higher-order binary MRF’s [14]. We provide approximation bounds sim-ilar to PD3 [19], and the method is fast in practice. It can optimize non-submodular MRF’s, and additionally can in-corporate problem-specific knowledge in the form of fusion proposals. We compare experimentally against the exist-ing approaches that can efficiently handle these difficult en-ergy functions [6, 10, 11]. For higher-order denoising and stereo MRF’s, we produce lower energy while running sig-nificantly faster. 1. Higher-order MRFs There is widespread interest in higher-order MRF’s for problems like denoising [23]and stereo [30], yet the result-ing energy functions have proven to be very difficult to min-imize. The optimization problem for a higher-order MRF is defined over a hypergraph with vertices V and cliques C plus a label set L. We minimize the cost of the labeling f: L|V | → < defined by f(x) =
Structured learning of sum-of-submodular higher order energy functions
, 1309
"... Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sum-of-submodular (SoS) functions [2, 17], which can be efficient ..."
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Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sum-of-submodular (SoS) functions [2, 17], which can be efficiently minimized via a variant of max flow called submodular flow [6]. SoS functions can naturally express higher order priors involving, e.g., local image patches; however, it is difficult to fully exploit their expressive power because they have so many parameters. Rather than trying to formulate existing higher order priors as an SoS function, we take a discriminative learning approach, effectively searching the space of SoS functions for a higher order prior that performs well on our training set. We adopt a structural SVM approach [14, 33] and formulate the training problem in terms of quadratic programming; as a result we can efficiently search the space of SoS priors via an extended cutting-plane algorithm. We also show how the state-of-the-art max flow method for vision problems [10] can be modified to efficiently solve the submodular flow problem. Experimental comparisons are made against the OpenCV implementation of the GrabCut interactive segmentation technique [27], which uses hand-tuned parameters instead of machine learning. On a standard dataset [11] our method learns higher order priors with hundreds of parameter values, and produces significantly better segmentations. While our focus is on binary labeling problems, we show that our techniques can be naturally generalized to handle more than two labels. 1.
Machine learning and convex optimization with Submodular Functions
- WORKSHOP ON COMBINATORIAL OPTIMIZATION- CARGESE, 2013
, 2013
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Parsimonious Labeling
"... We propose a new family of discrete energy minimiza-tion problems, which we call parsimonious labeling. Our energy function consists of unary potentials and high-order clique potentials. While the unary potentials are arbitrary, the clique potentials are proportional to the diversity of the set of u ..."
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We propose a new family of discrete energy minimiza-tion problems, which we call parsimonious labeling. Our energy function consists of unary potentials and high-order clique potentials. While the unary potentials are arbitrary, the clique potentials are proportional to the diversity of the set of unique labels assigned to the clique. Intuitively, our energy function encourages the labeling to be parsimo-nious, that is, use as few labels as possible. This in turn allows us to capture useful cues for important computer vi-sion applications such as stereo correspondence and image denoising. Furthermore, we propose an efficient graph-cuts based algorithm for the parsimonious labeling problem that provides strong theoretical guarantees on the quality of the solution. Our algorithm consists of three steps. First, we approximate a given diversity using a mixture of a novel hierarchical Pn Potts model. Second, we use a divide-and-conquer approach for each mixture component, where each subproblem is solved using an efficient α-expansion algo-rithm. This provides us with a small number of putative la-belings, one for each mixture component. Third, we choose the best putative labeling in terms of the energy value. Us-ing both synthetic and standard real datasets, we show that our algorithm significantly outperforms other graph-cuts based approaches. 1.
1Learning Weighted Lower Linear Envelope Potentials in Binary Markov Random Fields
"... Abstract—Markov random fields containing higher-order terms are becoming increasingly popular due to their ability to capture complicated relationships as soft constraints involving many output random variables. In computer vision an important class of constraints encode a preference for label consi ..."
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Abstract—Markov random fields containing higher-order terms are becoming increasingly popular due to their ability to capture complicated relationships as soft constraints involving many output random variables. In computer vision an important class of constraints encode a preference for label consistency over large sets of pixels and can be modeled using higher-order terms known as lower linear envelope potentials. In this paper we develop an algorithm for learning the parameters of binary Markov random fields with weighted lower linear envelope potentials. We first show how to perform exact energy minimization on these models in time polynomial in the number of variables and number of linear envelope functions. Then, with tractable inference in hand, we show how the parameters of the lower linear envelope potentials can be estimated from labeled training data within a max-margin learning framework. We explore three variants of the lower linear envelope parameterization and demonstrate results on both synthetic and real-world problems. Index Terms—higher-order MRFs, lower linear envelope potentials, max-margin learning 1
