346 research outputs found
Efficient Minimization of Decomposable Submodular Functions
Many combinatorial problems arising in machine learning can be reduced to the
problem of minimizing a submodular function. Submodular functions are a natural
discrete analog of convex functions, and can be minimized in strongly
polynomial time. Unfortunately, state-of-the-art algorithms for general
submodular minimization are intractable for larger problems. In this paper, we
introduce a novel subclass of submodular minimization problems that we call
decomposable. Decomposable submodular functions are those that can be
represented as sums of concave functions applied to modular functions. We
develop an algorithm, SLG, that can efficiently minimize decomposable
submodular functions with tens of thousands of variables. Our algorithm
exploits recent results in smoothed convex minimization. We apply SLG to
synthetic benchmarks and a joint classification-and-segmentation task, and show
that it outperforms the state-of-the-art general purpose submodular
minimization algorithms by several orders of magnitude.Comment: Expanded version of paper for Neural Information Processing Systems
201
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
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