Efficient Minimization of Higher Order Submodular Functions using Monotonic Boolean Functions

TL;DR

This paper introduces efficient algorithms for minimizing higher-order submodular functions by transforming them into quadratic functions using monotonic Boolean functions, reducing computational complexity in relevant applications.

Contribution

The work develops a novel linear programming approach to convert higher-order submodular functions into quadratic form with minimal auxiliary variables using monotonic Boolean functions.

Findings

Transformations enable efficient quadratic minimization of higher-order submodular functions.

Reduction in auxiliary variables leads to more compact and faster algorithms.

Applicable to subclasses of submodular functions in machine learning and computer vision.

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 $O(n^3 \log^2 n . E +n^4 {\log}^{O(1)} n)$ where $E$ is the time required to evaluate the function and $n$ 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) $k$ where $k \ll n$. 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. 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.