Results 1  10
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13
Submodularity beyond submodular energies: coupling edges in graph cuts
 IN CVPR
, 2011
"... We propose a new family of nonsubmodular global energy functions that still use submodularity internally to couple edges in a graph cut. We show it is possible to develop an efficient approximation algorithm that, thanks to the internal submodularity, can use standard graph cuts as a subroutine. We ..."
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Cited by 32 (17 self)
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We propose a new family of nonsubmodular global energy functions that still use submodularity internally to couple edges in a graph cut. We show it is possible to develop an efficient approximation algorithm that, thanks to the internal submodularity, can use standard graph cuts as a subroutine. We demonstrate the advantages of edge coupling in a natural setting, namely image segmentation. In particular, for finestructured objects and objects with shading variation, our structured edge coupling leads to significant improvements over standard approaches.
The Expressive Power of Binary Submodular Functions
, 2008
"... It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, arti ..."
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Cited by 15 (3 self)
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It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudoBoolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular functions of arity 4 can be expressed by binary submodular functions. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a socalled expressibility reduction to the MinCut problem. More importantly, our results imply limitations on this kind of reduction and establish for the first time that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.
On fast approximate submodular minimization
"... We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n 7) (O(n 5) oracle calls). We ..."
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Cited by 13 (6 self)
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We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n 7) (O(n 5) oracle calls). We therefore propose a fast approximate method to minimize arbitrary submodular functions. For a large subclass of submodular functions, the algorithm is exact. Other submodular functions are iteratively approximated by tight submodular upper bounds, and then repeatedly optimized. We show theoretical properties, and empirical results suggest significant speedups over minimum norm while retaining higher accuracies.
The expressibility of functions on the Boolean domain, with applications to Counting CSPs
"... An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post’s lattice gives a complete classification of ..."
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Cited by 8 (6 self)
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An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post’s lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of logsupermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial nonlsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function “implies”. As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexitytheoretic results, it is natural to ask whether the “implies” clone is equivalent to the clone of lsm functions. We use the Möbius transform and the Fourier transform to show that these
Minimizing a sum of submodular functions
 In Discrete Applied Mathematics
, 2012
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The Expressive Power of Valued Constraints: Hierarchies and Collapses
, 2008
"... In this paper, we investigate the ways in which a fixed collection of valued constraints can be combined to express other valued constraints. We show that in some cases, a large class of valued constraints, of all possible arities, can be expressed by using valued constraints over the same domain of ..."
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Cited by 4 (2 self)
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In this paper, we investigate the ways in which a fixed collection of valued constraints can be combined to express other valued constraints. We show that in some cases, a large class of valued constraints, of all possible arities, can be expressed by using valued constraints over the same domain of a fixed finite arity. We also show that some simple classes of valued constraints, including the set of all monotonic valued constraints with finite cost values, cannot be expressed by a subset of any fixed finite arity, and hence form an infinite hierarchy.
Cooperative cuts for image segmentation
, 2010
"... We propose a novel framework for graphbased cooperative regularization that uses submodular costs on graph edges. We introduce an efficient iterative algorithm to solve the resulting hard discrete optimization problem, and show that it has a guaranteed approximation factor. The edgesubmodular form ..."
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Cited by 2 (1 self)
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We propose a novel framework for graphbased cooperative regularization that uses submodular costs on graph edges. We introduce an efficient iterative algorithm to solve the resulting hard discrete optimization problem, and show that it has a guaranteed approximation factor. The edgesubmodular formulation is amenable to the same extensions as standard graph cut approaches, and applicable to a range of problems. We apply this method to the image segmentation problem. Specifically, Here, we apply it to introduce a discount for homogeneous boundaries in binary image segmentation on very difficult images, precisely, long thin objects and color and grayscale images with a shading gradient. The experiments show that significant portions of previously truncated objects are now preserved.
A note on some collapse results of valued constraints
, 2009
"... Valued constraint satisfaction problem (VCSP) is an optimisation framework originally coming from Artificial Intelligence and generalising the classical constraint satisfaction problem (CSP). The VCSP is powerful enough to describe many important classes of problems. In order to investigate the comp ..."
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Cited by 1 (0 self)
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Valued constraint satisfaction problem (VCSP) is an optimisation framework originally coming from Artificial Intelligence and generalising the classical constraint satisfaction problem (CSP). The VCSP is powerful enough to describe many important classes of problems. In order to investigate the complexity and expressive power of valued constraints, a number of algebraic tools have been developed in the literature. In this note we present alternative proofs of some known results without using the algebraic approach, but by representing valued constraints explicitly by combinations of other valued constraints.
Efficient Minimization of Higher Order Submodular functions using Monotonic Boolean Functions
, 2011
"... Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision and many others. The general solver has a complexity of O(n6 + n5L) where L is the time required to evaluate the function and n is the number of variables ..."
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Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision and many others. The general solver has a complexity of O(n6 + n5L) where L is the time required to evaluate the function and n is the number of variables [22]. On the other hand, many useful applications in computer vision and machine learning applications are defined over a special subclasses of submodular functions in which that can be written as the sum of many submodular cost functions defined over cliques containing few variables. In such functions, the pseudoBoolean (or polynomial) representation [2] of these subclasses are of degree (or order, or clique size) k where k muchlessthan n. In this work, we develop efficient algorithms for the minimization of this useful subclass of submodular functions. To do this, we define novel mapping that transform submodular functions of order k into quadratic ones, which can be efficiently minimized in O(n3) time using a maxflow algorithm. The underlying idea is to use auxiliary variables to model the higher order terms and the transformation is found using a carefully constructed linear program. In particular, we model the auxiliary variables as monotonic Boolean functions, allowing us to obtain a compact transformation using as few auxiliary variables as possible. Specifically, we show that our approach for fourth order function requires only 2 auxiliary variables in contrast to 30 or more variables used in existing approaches. In the general case, we give an upper bound for the number or auxiliary variables required to transform a function of order k using Dedekind number, which is substantially lower than the existing bound of 22k.