592 research outputs found
Getting Feasible Variable Estimates From Infeasible Ones: MRF Local Polytope Study
This paper proposes a method for construction of approximate feasible primal
solutions from dual ones for large-scale optimization problems possessing
certain separability properties. Whereas infeasible primal estimates can
typically be produced from (sub-)gradients of the dual function, it is often
not easy to project them to the primal feasible set, since the projection
itself has a complexity comparable to the complexity of the initial problem. We
propose an alternative efficient method to obtain feasibility and show that its
properties influencing the convergence to the optimum are similar to the
properties of the Euclidean projection. We apply our method to the local
polytope relaxation of inference problems for Markov Random Fields and
demonstrate its superiority over existing methods.Comment: 20 page, 4 figure
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Structured learning of sum-of-submodular higher order energy functions
Submodular functions can be exactly minimized in polynomial time, and the
special case that graph cuts solve with max flow \cite{KZ:PAMI04} has had
significant impact in computer vision
\cite{BVZ:PAMI01,Kwatra:SIGGRAPH03,Rother:GrabCut04}. In this paper we address
the important class of sum-of-submodular (SoS) functions
\cite{Arora:ECCV12,Kolmogorov:DAM12}, which can be efficiently minimized via a
variant of max flow called submodular flow \cite{Edmonds:ADM77}. 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
\cite{Joachims/etal/09a,Tsochantaridis/etal/04} 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
\cite{Goldberg:ESA11} can be modified to efficiently solve the submodular flow
problem. Experimental comparisons are made against the OpenCV implementation of
the GrabCut interactive segmentation technique \cite{Rother:GrabCut04}, which
uses hand-tuned parameters instead of machine learning. On a standard dataset
\cite{Gulshan:CVPR10} 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
Combinatorial persistency criteria for multicut and max-cut
In combinatorial optimization, partial variable assignments are called
persistent if they agree with some optimal solution. We propose persistency
criteria for the multicut and max-cut problem as well as fast combinatorial
routines to verify them. The criteria that we derive are based on mappings that
improve feasible multicuts, respectively cuts. Our elementary criteria can be
checked enumeratively. The more advanced ones rely on fast algorithms for upper
and lower bounds for the respective cut problems and max-flow techniques for
auxiliary min-cut problems. Our methods can be used as a preprocessing
technique for reducing problem sizes or for computing partial optimality
guarantees for solutions output by heuristic solvers. We show the efficacy of
our methods on instances of both problems from computer vision, biomedical
image analysis and statistical physics
Full Flow: Optical Flow Estimation By Global Optimization over Regular Grids
We present a global optimization approach to optical flow estimation. The
approach optimizes a classical optical flow objective over the full space of
mappings between discrete grids. No descriptor matching is used. The highly
regular structure of the space of mappings enables optimizations that reduce
the computational complexity of the algorithm's inner loop from quadratic to
linear and support efficient matching of tens of thousands of nodes to tens of
thousands of displacements. We show that one-shot global optimization of a
classical Horn-Schunck-type objective over regular grids at a single resolution
is sufficient to initialize continuous interpolation and achieve
state-of-the-art performance on challenging modern benchmarks.Comment: To be presented at CVPR 201
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
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