2,270 research outputs found
Rounding-based Moves for Semi-Metric Labeling
International audienceSemi-metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given semi-metric distance function over the label set. Popular methods for solving semi-metric labeling include (i) move-making algorithms, which iteratively solve a minimum st-cut problem; and (ii) the linear programming (LP) relaxation based approach. In order to convert the fractional solution of the LP relaxation to an integer solution, several randomized rounding procedures have been developed in the literature. We consider a large class of parallel rounding procedures, and design move-making algorithms that closely mimic them. We prove that the multiplicative bound of a move-making algorithm exactly matches the approximation factor of the corresponding rounding procedure for any arbitrary distance function. Our analysis includes all known results for move-making algorithms as special cases
Parsimonious Labeling
We propose a new family of discrete energy minimization problems, which we
call parsimonious labeling. Specifically, our energy functional consists of
unary potentials and high-order clique potentials. While the unary potentials
are arbitrary, the clique potentials are proportional to the {\em diversity} of
set of the unique labels assigned to the clique. Intuitively, our energy
functional encourages the labeling to be parsimonious, that is, use as few
labels as possible. This in turn allows us to capture useful cues for important
computer vision 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
Potts model. Second, we use a divide-and-conquer approach for each mixture
component, where each subproblem is solved using an effficient
-expansion algorithm. This provides us with a small number of putative
labelings, one for each mixture component. Third, we choose the best putative
labeling in terms of the energy value. Using both sythetic and standard real
datasets, we show that our algorithm significantly outperforms other graph-cuts
based approaches
Local Guarantees in Graph Cuts and Clustering
Correlation Clustering is an elegant model that captures fundamental graph
cut problems such as Min Cut, Multiway Cut, and Multicut, extensively
studied in combinatorial optimization. Here, we are given a graph with edges
labeled or and the goal is to produce a clustering that agrees with the
labels as much as possible: edges within clusters and edges across
clusters. The classical approach towards Correlation Clustering (and other
graph cut problems) is to optimize a global objective. We depart from this and
study local objectives: minimizing the maximum number of disagreements for
edges incident on a single node, and the analogous max min agreements
objective. This naturally gives rise to a family of basic min-max graph cut
problems. A prototypical representative is Min Max Cut: find an cut
minimizing the largest number of cut edges incident on any node. We present the
following results: an -approximation for the problem of
minimizing the maximum total weight of disagreement edges incident on any node
(thus providing the first known approximation for the above family of min-max
graph cut problems), a remarkably simple -approximation for minimizing
local disagreements in complete graphs (improving upon the previous best known
approximation of ), and a -approximation for
maximizing the minimum total weight of agreement edges incident on any node,
hence improving upon the -approximation that follows from
the study of approximate pure Nash equilibria in cut and party affiliation
games
GRMA: Generalized Range Move Algorithms for the efficient optimization of MRFs
Markov Random Fields (MRF) have become an
important tool for many vision applications, and the optimization
of MRFs is a problem of fundamental importance.
Recently, Veksler and Kumar et al. proposed the range move
algorithms, which are some of the most successful optimizers.
Instead of considering only two labels as in previous
move-making algorithms, they explore a large search space
over a range of labels in each iteration, and significantly
outperform previous move-making algorithms. However, two
problems have greatly limited the applicability of range
move algorithms: 1) They are limited in the energy functions
they can handle (i.e., only truncated convex functions); 2)
They tend to be very slow compared to other move-making
algorithms (e.g., �-expansion and ��-swap). In this paper,
we propose two generalized range move algorithms (GRMA)
for the efficient optimization of MRFs. To address the
first problem, we extend the GRMAs to more general energy
functions by restricting the chosen labels in each move so
that the energy function is submodular on the chosen subset.
Furthermore, we provide a feasible sufficient condition for
choosing these subsets of labels. To address the second
problem, we dynamically obtain the iterative moves by solving
set cover problems. This greatly reduces the number of
moves during the optimization.We also propose a fast graph
construction method for the GRMAs. Experiments show
that the GRMAs offer a great speedup over previous range
move algorithms, while yielding competitive solutions
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