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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 ..."
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Cited by 25 (3 self)
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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. 1
Fast Semidifferential-based Submodular Function Optimization
, 2013
"... 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, off ..."
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Cited by 14 (3 self)
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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 minimization 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.
Cooperative Cuts: Graph Cuts with Submodular Edge Weights
, 2010
"... We introduce a problem we call Cooperative cut, where the goal is to find a minimum-cost graph cut but where a submodular function is used to define the cost of a subsets of edges. That means, the cost of an edge that is added to the current cut set C depends on the edges in C. This generalization o ..."
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Cited by 11 (8 self)
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We introduce a problem we call Cooperative cut, where the goal is to find a minimum-cost graph cut but where a submodular function is used to define the cost of a subsets of edges. That means, the cost of an edge that is added to the current cut set C depends on the edges in C. This generalization of the cost in the standard min-cut problem to a submodular cost function immediately makes the problem harder. Not only do we prove NP hardness even for nonnegative submodular costs, but also show a lower bound of Ω(|V | 1/3) on the approximation factor for the problem. On the positive side, we propose and compare four approximation algorithms with an overall approximation factor of min { |V |/2, |C ∗ |, O ( √ |E | log |V |), |Pmax | } , where C ∗ is the optimal solution, and Pmax is the longest s, t path across the cut between given s, t. We also introduce additional heuristics for the problem which have attractive properties from the perspective of practical applications and implementations in that existing fast min-cut libraries may be used as subroutines. Both our approximation algorithms, and our heuristics, appear to do well in practice.
Weighted Universal Recovery, Practical Secrecy, and an Efficient Algorithm for Solving Both
- In Communication, Control, and Computing (Allerton), 2011 49th Annual Allerton Conference on
, 2011
"... Abstract—In this paper, we consider a network of n nodes, each initially possessing a subset of packets. Each node is permitted to broadcast functions of its own packets and the messages it receives to all other nodes via an error-free channel. We provide an algorithm that efficiently solves the Wei ..."
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Abstract—In this paper, we consider a network of n nodes, each initially possessing a subset of packets. Each node is permitted to broadcast functions of its own packets and the messages it receives to all other nodes via an error-free channel. We provide an algorithm that efficiently solves the Weighted Universal Recovery Problem and the Secrecy Generation Problem for this network. In the Weighted Universal Recovery Problem, the goal is to design a sequence of transmissions that ultimately permits all nodes to recover all packets initially present in the network. We show how to compute a transmission scheme that is optimal in the sense that the weighted sum of the number of transmissions is minimized. For the Secrecy Generation Problem, the goal is to generate a secret-key among the nodes that cannot be derived by an eavesdropper privy to the transmissions. In particular, we wish to generate a secret-key of maximum size. Further, we discuss private-key generation, which applies to the case where a subset of nodes is compromised by the eavesdropper. For the network under consideration, both of these problems are combinatorial in nature. We demonstrate that each of these problems can be solved efficiently and exactly. Notably, we do not require any terms to grow asymptotically large to obtain our results. This is in sharp contrast to classical information-theoretic problems despite the fact that our problems are informationtheoretic in nature. Finally, the algorithm we describe efficiently solves an Integer Linear Program of a particular form. Due to the general form we consider, it may prove useful beyond these applications. I.
Sensor planning for a symbiotic UAV and UGV system for precision agriculture
- In Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems
, 2013
"... Abstract-We study the problem of coordinating an Unmanned Aerial Vehicle (UAV) and Unmanned Ground Vehicle (UGV) to collect data for a precision agriculture application. The ground and aerial measurements collected by the system are used for estimating Nitrogen (N) levels across a farm field. These ..."
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Abstract-We study the problem of coordinating an Unmanned Aerial Vehicle (UAV) and Unmanned Ground Vehicle (UGV) to collect data for a precision agriculture application. The ground and aerial measurements collected by the system are used for estimating Nitrogen (N) levels across a farm field. These estimates in turn guide fertilizer application. The capability to apply the right amount of fertilizer at the right time can drastically reduce fertilizer usage which is desirable from an environmental and economic standpoint. We propose to use a symbiotic UAV and UGV system in which the UGV is capable of muling the UAV to various deployment locations. This would allow the system to overcome the short battery life of a typical UAV. Our goal is to estimate N levels over the field and assign each point in the field into classes indicating N-deficiency levels. Towards building such a system, the paper makes the following contributions: First, we present a method to identify points whose probability of being misclassified is above a threshold, termed as Potentially Mislabeled (PML). Second, we study the problem of planning the UAV path to visit the maximum number of PML points subject to its energy budget. The novelty of our formulation is the capability of the UGV to mule the UAV to deployment points. Third, we introduce a new path planning problem in which the UGV must take a measurement near each PML point visited by the UAV. The goal is to minimize the total time spent in traveling and taking measurements. For both problems, we present constant-factor approximation algorithms. Finally, we demonstrate the utility of the system and our algorithms with simulations which use manually collected data from the field as well as realistic energy models for the UAV and the UGV.
Branch and Bound Strategies for Non-maximal Suppression in Object Detection
"... Abstract. In this work, we are concerned with the detection of multiple objects in an image. We demonstrate that typically applied objectives have the structure of a random field model, but that the energies resulting from non-maximal suppression terms lead to the maximization of a submodular functi ..."
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Abstract. In this work, we are concerned with the detection of multiple objects in an image. We demonstrate that typically applied objectives have the structure of a random field model, but that the energies resulting from non-maximal suppression terms lead to the maximization of a submodular function. This is in general a difficult problem to solve, which is made worse by the very large size of the output space. We make use of an optimal approximation result for this form of problem by employing a greedy algorithm that finds one detection at a time. We show that we can adopt a branch-and-bound strategy that efficiently explores the space of all subwindows to optimally detect single objects while incorporating pairwise energies resulting from previous detections. This leads to a series of inter-related branch-and-bound optimizations, which we characterize by several new theoretical results. We then show empirically that optimal branch-and-bound efficiency gains can be achieved by a simple strategy of reusing priority queues from previous detections, resulting in speedups of up to a factor of three on the PASCAL VOC data set as compared with serial application of branch-and-bound. 1
Fast Flux Discriminant for Large-Scale Sparse Nonlinear Classification
"... In this paper, we propose a novel supervised learning method, Fast Flux Discriminant (FFD), for large-scale nonlinear clas-sification. Compared with other existing methods, FFD has unmatched advantages, as it attains the efficiency and in-terpretability of linear models as well as the accuracy of no ..."
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In this paper, we propose a novel supervised learning method, Fast Flux Discriminant (FFD), for large-scale nonlinear clas-sification. Compared with other existing methods, FFD has unmatched advantages, as it attains the efficiency and in-terpretability of linear models as well as the accuracy of nonlinear models. It is also sparse and naturally handles mixed data types. It works by decomposing the kernel den-sity estimation in the entire feature space into selected low-dimensional subspaces. Since there are many possible sub-spaces, we propose a submodular optimization framework for subspace selection. The selected subspace predictions are then transformed to new features on which a linear model can be learned. Besides, since the transformed features nat-urally expect non-negative weights, we only require smooth optimization even with the `1 regularization. Unlike other nonlinear models such as kernel methods, the FFD model is interpretable as it gives importance weights on the original features. Its training and testing are also much faster than traditional kernel models. We carry out extensive empiri-cal studies on real-world datasets and show that the pro-posed model achieves state-of-the-art classification results with sparsity, interpretability, and exceptional scalability. Our model can be learned in minutes on datasets with mil-lions of samples, for which most existing nonlinear methods will be prohibitively expensive in space and time.
Notes on graph cuts with submodular edge weights
- In Workshop on Discrete Optimization in Machine Learning: Submodularity, Sparsity & Polyhedra (DISCML
, 2009
"... Generalizing the cost in the standard min-cut problem to a submodular cost function immediately makes the problem harder. Not only do we prove NP hardness even for nonnegative submodular costs, but also show a lower bound of Ω(|V | 1/3) on the approximation factor for the (s, t) cut version of the p ..."
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Cited by 1 (0 self)
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Generalizing the cost in the standard min-cut problem to a submodular cost function immediately makes the problem harder. Not only do we prove NP hardness even for nonnegative submodular costs, but also show a lower bound of Ω(|V | 1/3) on the approximation factor for the (s, t) cut version of the problem. On the positive side, we propose and compare three approximation algorithms with an overall approximation factor of O(min{|V |, √ |E | log |V |}) that appear to do well in practice. 1
