80 research outputs found
The power of linear programming for general-valued CSPs
Let , called the domain, be a fixed finite set and let , called
the valued constraint language, be a fixed set of functions of the form
, where different functions might have
different arity . We study the valued constraint satisfaction problem
parametrised by , denoted by VCSP. These are minimisation
problems given by variables and the objective function given by a sum of
functions from , each depending on a subset of the variables.
Finite-valued constraint languages contain functions that take on only rational
values and not infinite values.
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation (BLP). For a valued constraint language , BLP is a decision
procedure for if and only if admits a symmetric fractional
polymorphism of every arity. For a finite-valued constraint language ,
BLP is a decision procedure if and only if admits a symmetric
fractional polymorphism of some arity, or equivalently, if admits a
symmetric fractional polymorphism of arity 2.
Using these results, we obtain tractability of several novel classes of
problems, including problems over valued constraint languages that are: (1)
submodular on arbitrary lattices; (2) -submodular on arbitrary finite
domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors
(arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213)
to appear in SIAM Journal on Computing (SICOMP
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
Maximum Persistency via Iterative Relaxed Inference with Graphical Models
We consider the NP-hard problem of MAP-inference for undirected discrete
graphical models. We propose a polynomial time and practically efficient
algorithm for finding a part of its optimal solution. Specifically, our
algorithm marks some labels of the considered graphical model either as (i)
optimal, meaning that they belong to all optimal solutions of the inference
problem; (ii) non-optimal if they provably do not belong to any solution. With
access to an exact solver of a linear programming relaxation to the
MAP-inference problem, our algorithm marks the maximal possible (in a specified
sense) number of labels. We also present a version of the algorithm, which has
access to a suboptimal dual solver only and still can ensure the
(non-)optimality for the marked labels, although the overall number of the
marked labels may decrease. We propose an efficient implementation, which runs
in time comparable to a single run of a suboptimal dual solver. Our method is
well-scalable and shows state-of-the-art results on computational benchmarks
from machine learning and computer vision.Comment: Reworked version, submitted to PAM
Discrete graphical models -- an optimization perspective
This monograph is about discrete energy minimization for discrete graphical
models. It considers graphical models, or, more precisely, maximum a posteriori
inference for graphical models, purely as a combinatorial optimization problem.
Modeling, applications, probabilistic interpretations and many other aspects
are either ignored here or find their place in examples and remarks only. It
covers the integer linear programming formulation of the problem as well as its
linear programming, Lagrange and Lagrange decomposition-based relaxations. In
particular, it provides a detailed analysis of the polynomially solvable
acyclic and submodular problems, along with the corresponding exact
optimization methods. Major approximate methods, such as message passing and
graph cut techniques are also described and analyzed comprehensively. The
monograph can be useful for undergraduate and graduate students studying
optimization or graphical models, as well as for experts in optimization who
want to have a look into graphical models. To make the monograph suitable for
both categories of readers we explicitly separate the mathematical optimization
background chapters from those specific to graphical models.Comment: 270 page
Higher-Order Regularization in Computer Vision
At the core of many computer vision models lies the minimization of an objective function consisting of a sum of functions with few arguments. The order of the objective function is defined as the highest number of arguments of any summand. To reduce ambiguity and noise in the solution, regularization terms are included into the objective function, enforcing different properties of the solution. The most commonly used regularization is penalization of boundary length, which requires a second-order objective function. Most of this thesis is devoted to introducing higher-order regularization terms and presenting efficient minimization schemes. One of the topics of the thesis covers a reformulation of a large class of discrete functions into an equivalent form. The reformulation is shown, both in theory and practical experiments, to be advantageous for higher-order regularization models based on curvature and second-order derivatives. Another topic is the parametric max-flow problem. An analysis is given, showing its inherent limitations for large-scale problems which are common in computer vision. The thesis also introduces a segmentation approach for finding thin and elongated structures in 3D volumes. Using a line-graph formulation, it is shown how to efficiently regularize with respect to higher-order differential geometric properties such as curvature and torsion. Furthermore, an efficient optimization approach for a multi-region model is presented which, in addition to standard regularization, is able to enforce geometric constraints such as inclusion or exclusion of different regions. The final part of the thesis deals with dense stereo estimation. A new regularization model is introduced, penalizing the second-order derivatives of a depth or disparity map. Compared to previous second-order approaches to dense stereo estimation, the new regularization model is shown to be more easily optimized
Advances in Graph-Cut Optimization: Multi-Surface Models, Label Costs, and Hierarchical Costs
Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of low-level inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energies let us state each problem as a clear, precise objective function. Minimizing the correct energy would, hypothetically, yield a good solution to the corresponding problem. Unfortunately, even for low-level problems we are confronted by energies that are computationally hard—often NP-hard—to minimize. As a consequence, a rather large portion of computer vision research is dedicated to proposing better energies and better algorithms for energies. This dissertation presents work along the same line, specifically new energies and algorithms based on graph cuts.
We present three distinct contributions. First we consider biomedical segmentation where the object of interest comprises multiple distinct regions of uncertain shape (e.g. blood vessels, airways, bone tissue). We show that this common yet difficult scenario can be modeled as an energy over multiple interacting surfaces, and can be globally optimized by a single graph cut. Second, we introduce multi-label energies with label costs and provide algorithms to minimize them. We show how label costs are useful for clustering and robust estimation problems in vision. Third, we characterize a class of energies with hierarchical costs and propose a novel hierarchical fusion algorithm with improved approximation guarantees. Hierarchical costs are natural for modeling an array of difficult problems, e.g. segmentation with hierarchical context, simultaneous estimation of motions and homographies, or detecting hierarchies of patterns
Efficient Minimization of Higher Order Submodular Functions using Monotonic Boolean Functions
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 where is the time required to evaluate the function and
is the number of variables \cite{Lee2015}. On the other hand, many computer
vision and machine learning problems are defined over special subclasses of
submodular functions that can be written as the sum of many submodular cost
functions defined over cliques containing few variables. In such functions, the
pseudo-Boolean (or polynomial) representation \cite{BorosH02} of these
subclasses are of degree (or order, or clique size) where . 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 into quadratic ones. 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
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