Quadratic pseudo-Boolean optimization: Difference between revisions

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{{short description|Combinatorial optimization method for pseudo-Boolean functions}}

'''Quadratic pseudo-Boolean optimisation''' ('''QPBO''') is a [[combinatorial optimization]] method for minimizing quadratic [[pseudo-Boolean function]]s in the form

:<math> f(\mathbf{x}) = w_0 + \sum_{p \in V} w_p(x_p) + \sum_{(p, q) \in E} w_{pq}(x_p, x_q) </math>

in the [[binary data|binary variables]] <math>x_p \in \{0, 1\} \; \forall p \in V = \{1, \dots, n\}</math>, with <math>E \subseteq V \times V</math>. If <math>f</math> is [[Pseudo-Boolean_function#Submodularity|submodular]] then QPBO produces a global optimum equivalently to [[graph cut optimization]], while if <math>f</math> contains non-submodular terms then the algorithm produces a partial solution with specific optimality properties, in both cases in [[polynomial time]].<ref name="review" />

QPBO is a useful tool for inference on [[Markov random field]]s and [[conditional random field]]s, and has applications in [[computer vision]] problems such as [[image segmentation]] and [[stereo cameras|stereo matching]].<ref name="rother" />

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== Higher order terms ==

The problem of optimizing higher-order pseudo-boolean functions is generally difficult.{{cn|date=June 2021}} It is always possible to reduce a higher-order function to a quadratic function which is equivalent with respect to the optimisation, problem known as "higher-order [[clique (graph theory)|clique]] reduction" (HOCR), and the result of such reduction can be optimized with QPBO. Generic methods for reduction of arbitrary functions rely on specific substitution rules and in the general case they require the introduction of auxiliary variables.<ref name="fix" /> In practice most terms can be reduced without introducing additional variables, resulting in a simpler optimization problem, and the remaining terms can be reduced exactly, with addition of auxiliary variables, or approximately, without addition of any new variable.<ref name="elc" />

==Notes==

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== Notes ==

<references group="note">

<ref name="normal form">The representation of a pseudo-Boolean function with coefficients <math>\mathbf{w} = (w_0, w_1, \dots, w_{nn})</math> is not unique, and if two coefficient vectors <math>\mathbf{w}</math> and <math>\mathbf{w}'</math> represent the same function then <math>\mathbf{w}'</math> is said to be a reparametrisation of <math>\mathbf{w}</math> and vice versa. In some constructions it is useful to ensure that the function has a specific form, called ''noramlnormal form'', which is always defined for any function, and it is not unique. A function <math>f</math> is in normal form if the two following conditions hold (Kolmogorov and Rother (2007)):

# <math>\min \{ w_p^0, w_p^1 \} = 0</math> for each <math>p \in V</math>;

# <math>\min \{ w_{pq}^{0j}, w_{pq}^{1j} \} = 0</math> for each <math>(p, q) \in E</math> and for each <math>j \in \{0, 1\}</math>.