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Efficient Minimization of Higher Order Submodular Functions using Monotonic Boolean Functions.

Tue, 09/13/2011 - 15:01 — SomNambul
Authors: Srikumar Ramalingam, Chris Russell, Lubor Ladicky, Philip H.S. Torr
Subjects: Data Structures and Algorithms
link: http://arxiv.org/abs/1109.2304
Abstract

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 $O(n^6+n^5L)$ where $L$ is
the time required to evaluate the function and $n$ is the number of variables
\cite{orlin09}. On the other hand, many useful applications in computer vision
and machine learning applications are defined over a special subclasses of
submodular functions in which 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) $k$ where $k<<n$. 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 $k$ into quadratic ones, which can be
efficiently minimized in $O(n^3)$ time using a max-flow algorithm. 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. Specifically, we show that our approach for fourth order function
requires only 2 auxiliary variables in contrast to 30 or more variables used in
existing approaches. In the general case, we give an upper bound for the number
or auxiliary variables required to transform a function of order $k$ using
Dedekind number, which is substantially lower than the existing bound of
$2^{2^k}$.

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