Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow KZ:PAMI04 has had significant impact in computer vision BVZ:PAMI01,Kwatra:SIGGRAPH03,Rother:GrabCut04 . In this paper we address the important class of sum-of-submodular (SoS) functions Arora:ECCV12,Kolmogorov:DAM12 , which can be efficiently minimized via a variant of max flow called submodular flow 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 Joachims et al 09a,Tsochantaridis et al 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 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 Rother:GrabCut04 , which uses hand-tuned parameters instead of machine learning. On a standard dataset 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.
In this paper, we will learn submodular discriminant functions. Prior work on learning submodular functions falls into three categories: submodular function regression @cite_11 , maximization of submodular discriminant functions, and minimization of submodular discriminant functions.
@cite_11
2766474417 1601868537 2951977866 2545520746
We investigate the existence of approximation algorithms for maximization of submodular functions, that run in a fixed parameter tractable (FPT) time. Given a non-decreasing submodular set function v:2 X →R v : 2 X → R the goal is to select a subset S of K elements from X such that v(S) v ( S ) is maximized. We identify three properties of set functions, referred to as p -separability properties, and we argue that many real-life problems can be expressed as maximization of submodular, p -separable functions, with low values of the parameter p . We present FPT approximation schemes for the minimization and maximization variants of the problem, for several parameters that depend on characteristics of the optimized set function, such as p and K . Symmetric submodular functions are an important family of submodular functions capturing many interesting cases, including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little attention by current research, unlike similar minimization problems that have been widely studied. In this work, we identify a few submodular maximization problems for which one can get a better approximation for symmetric objectives than the state-of-the-art approximation for general submodular functions. We first consider the problem of maximizing a non-negative symmetric submodular function f:2N → R+ subject to a down-monotone solvable polytope P ⊆ [0, 1]N. For this problem, we describe an algorithm producing a fractional solution of value at least 0.432 ċ f(OPT), where OPT is the optimal integral solution. Our second result considers the problem max f(S)c vSv e k for a non-negative symmetric submodular function f:2N → R+. For this problem, we give an approximation ratio that depends on the value k vNv and is always at least 0.432. Our method can also be applied to non-negative non-symmetric submodular functions, in which case it produces 1 e − o(1) approximation, improving over the best-known result for this problem. For unconstrained maximization of a non-negative symmetric submodular function, we describe a deterministic linear-time 1 2-approximation algorithm. Finally, we give a [1 − (1 − 1 k)k − 1]-approximation algorithm for Submodular Welfare with k players having identical non-negative submodular utility functions and show that this is the best possible approximation ratio for the problem. We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (sub- and super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes steps towards providing a unifying paradigm applicable to both submodular min- imization and maximization, problems that historically have been treated quite distinctly. The practicality of our algorithms is important since interest in submodularity, owing to its natural and wide applicability, has recently been in ascendance within machine learning. We analyze theoretical properties of our algorithms for minimization and maximization, and show that many state-of-the-art maximization algorithms are special cases. Lastly, we complement our theoretical analyses with supporting empirical experiments. Submodularity is one of the most important properties of combinatorial optimization, and k-submodularity is a generalization of submodularity. Maximization of a k-submodular function is NP-hard, and approximation algorithms have been studied. Most of algorithms use randomization and achieve the approximation ratio as the expected value. For unconstrained submodular maximization, [Buchbinder and Feldman 2016] gave a derandomization scheme, and showed that randomness is not necessary for obtaining optimal results. In this paper, we extend their scheme for unconstrained k-submodular maximization and give two deterministic algorithms for k-submodular maximization. The first is for monotone k-submodular functions. We derandomize (k (2k-1) )-approximation algorithm [Iwata, Tanigawa, and Yoshida 2016], and achieve the same approximation ratio. The second is for nonmonotone functions with (k 2 ). It achieves (k (3k-2) )-approximation, and it is better than a simple greedy algorithm [Ward and Živný 2016] which achieves 1 3-approximation for (k 3 ).