27 research outputs found
Structured learning of sum-of-submodular higher order energy functions
Submodular functions can be exactly minimized in polynomial time, and the
special case that graph cuts solve with max flow \cite{KZ:PAMI04} has had
significant impact in computer vision
\cite{BVZ:PAMI01,Kwatra:SIGGRAPH03,Rother:GrabCut04}. In this paper we address
the important class of sum-of-submodular (SoS) functions
\cite{Arora:ECCV12,Kolmogorov:DAM12}, which can be efficiently minimized via a
variant of max flow called submodular flow \cite{Edmonds:ADM77}. SoS functions
can naturally express higher order priors involving, e.g., local image patches;
however, it is difficult to fully exploit their expressive power because they
have so many parameters. Rather than trying to formulate existing higher order
priors as an SoS function, we take a discriminative learning approach,
effectively searching the space of SoS functions for a higher order prior that
performs well on our training set. We adopt a structural SVM approach
\cite{Joachims/etal/09a,Tsochantaridis/etal/04} and formulate the training
problem in terms of quadratic programming; as a result we can efficiently
search the space of SoS priors via an extended cutting-plane algorithm. We also
show how the state-of-the-art max flow method for vision problems
\cite{Goldberg:ESA11} can be modified to efficiently solve the submodular flow
problem. Experimental comparisons are made against the OpenCV implementation of
the GrabCut interactive segmentation technique \cite{Rother:GrabCut04}, which
uses hand-tuned parameters instead of machine learning. On a standard dataset
\cite{Gulshan:CVPR10} our method learns higher order priors with hundreds of
parameter values, and produces significantly better segmentations. While our
focus is on binary labeling problems, we show that our techniques can be
naturally generalized to handle more than two labels
Structured learning of sum-of-submodular higher order energy functions
Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sum-of-submodular (SoS) functions [2, 17], which can be efficiently minimized via a variant of max flow called submodular flow [6]. SoS functions can naturally express higher order priors involving, e.g., local image patches; however, it is difficult to fully exploit their expressive power because they have so many parameters. Rather than trying to formulate existing higher order priors as an SoS function, we take a discriminative learning approach, effectively searching the space of SoS functions for a higher order prior that performs well on our training set. We adopt a structural SVM approach [14, 33] and formulate the training problem in terms of quadratic programming; as a result we can efficiently search the space of SoS priors via an extended cutting-plane algorithm. We also show how the state-of-the-art max flow method for vision problems [10] can be modified to efficiently solve the submodular flow problem. Experimental comparisons are made against the OpenCV implementation of the GrabCut interactive segmentation technique [27], which uses hand-tuned parameters instead of machine learning. On a standard dataset [11] our method learns higher order priors with hundreds of parameter values, and produces significantly better segmentations. While our focus is on binary labeling problems, we show that our techniques can be naturally generalized to handle more than two labels. 1
The Limitations of Optimization from Samples
In this paper we consider the following question: can we optimize objective
functions from the training data we use to learn them? We formalize this
question through a novel framework we call optimization from samples (OPS). In
OPS, we are given sampled values of a function drawn from some distribution and
the objective is to optimize the function under some constraint.
While there are interesting classes of functions that can be optimized from
samples, our main result is an impossibility. We show that there are classes of
functions which are statistically learnable and optimizable, but for which no
reasonable approximation for optimization from samples is achievable. In
particular, our main result shows that there is no constant factor
approximation for maximizing coverage functions under a cardinality constraint
using polynomially-many samples drawn from any distribution.
We also show tight approximation guarantees for maximization under a
cardinality constraint of several interesting classes of functions including
unit-demand, additive, and general monotone submodular functions, as well as a
constant factor approximation for monotone submodular functions with bounded
curvature