31 research outputs found
Minimizing a sum of submodular functions
We consider the problem of minimizing a function represented as a sum of
submodular terms. We assume each term allows an efficient computation of {\em
exchange capacities}. This holds, for example, for terms depending on a small
number of variables, or for certain cardinality-dependent terms.
A naive application of submodular minimization algorithms would not exploit
the existence of specialized exchange capacity subroutines for individual
terms. To overcome this, we cast the problem as a {\em submodular flow} (SF)
problem in an auxiliary graph, and show that applying most existing SF
algorithms would rely only on these subroutines.
We then explore in more detail Iwata's capacity scaling approach for
submodular flows (Math. Programming, 76(2):299--308, 1997). In particular, we
show how to improve its complexity in the case when the function contains
cardinality-dependent terms.Comment: accepted to "Discrete Applied Mathematics
Random Coordinate Descent Methods for Minimizing Decomposable Submodular Functions
Submodular function minimization is a fundamental optimization problem that
arises in several applications in machine learning and computer vision. The
problem is known to be solvable in polynomial time, but general purpose
algorithms have high running times and are unsuitable for large-scale problems.
Recent work have used convex optimization techniques to obtain very practical
algorithms for minimizing functions that are sums of ``simple" functions. In
this paper, we use random coordinate descent methods to obtain algorithms with
faster linear convergence rates and cheaper iteration costs. Compared to
alternating projection methods, our algorithms do not rely on full-dimensional
vector operations and they converge in significantly fewer iterations
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
Generalized roof duality and bisubmodular functions
Consider a convex relaxation of a pseudo-boolean function . We
say that the relaxation is {\em totally half-integral} if is a
polyhedral function with half-integral extreme points , and this property is
preserved after adding an arbitrary combination of constraints of the form
, , and where \gamma\in\{0, 1, 1/2} is a
constant. A well-known example is the {\em roof duality} relaxation for
quadratic pseudo-boolean functions . 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 by
establishing a one-to-one correspondence with {\em 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.Comment: 14 pages. Shorter version to appear in NIPS 201
On the Convergence Rate of Decomposable Submodular Function Minimization
Submodular functions describe a variety of discrete problems in machine
learning, signal processing, and computer vision. However, minimizing
submodular functions poses a number of algorithmic challenges. Recent work
introduced an easy-to-use, parallelizable algorithm for minimizing submodular
functions that decompose as the sum of "simple" submodular functions.
Empirically, this algorithm performs extremely well, but no theoretical
analysis was given. In this paper, we show that the algorithm converges
linearly, and we provide upper and lower bounds on the rate of convergence. Our
proof relies on the geometry of submodular polyhedra and draws on results from
spectral graph theory.Comment: 17 pages, 3 figure
Reflection methods for user-friendly submodular optimization
Recently, it has become evident that submodularity naturally captures widely
occurring concepts in machine learning, signal processing and computer vision.
Consequently, there is need for efficient optimization procedures for
submodular functions, especially for minimization problems. While general
submodular minimization is challenging, we propose a new method that exploits
existing decomposability of submodular functions. In contrast to previous
approaches, our method is neither approximate, nor impractical, nor does it
need any cumbersome parameter 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 reconstruction. In our experiments, we
illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013
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