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14
Learning Mixtures of Submodular Functions for Image Collection Summarization
"... We address the problem of image collection summarization by learning mixtures of submodular functions. Submodularity is useful for this problem since it naturally represents characteristics such as fidelity and diversity, desirable for any summary. Several previously proposed image summarization sco ..."
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We address the problem of image collection summarization by learning mixtures of submodular functions. Submodularity is useful for this problem since it naturally represents characteristics such as fidelity and diversity, desirable for any summary. Several previously proposed image summarization scoring methodologies, in fact, instinctively arrived at submodularity. We provide classes of submodular component functions (including some which are instantiated via a deep neural network) over which mixtures may be learnt. We formulate the learning of such mixtures as a supervised problem via largemargin structured prediction. As a loss function, and for automatic summary scoring, we introduce a novel summary evaluation method called VROUGE, and test both submodular and nonsubmodular optimization (using the submodularsupermodular procedure) to learn a mixture of submodular functions. Interestingly, using nonsubmodular optimization to learn submodular functions provides the best results. We also provide a new data set consisting of 14 realworld image collections along with many humangenerated ground truth summaries collected using Amazon Mechanical Turk. We compare our method with previous work on this problem and show that our learning approach outperforms all competitors on this new data set. This paper provides, to our knowledge, the first systematic approach for quantifying the problem of image collection summarization, along with a new data set of image collections and human summaries. 1
From MAP to marginals: Variational inference in Bayesian submodular models
 In Neural Information Processing Systems (NIPS
, 2014
"... Submodular optimization has found many applications in machine learning and beyond. We carry out the first systematic investigation of inference in probabilistic models defined through submodular functions, generalizing regular pairwise MRFs and Determinantal Point Processes. In particular, we pres ..."
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Submodular optimization has found many applications in machine learning and beyond. We carry out the first systematic investigation of inference in probabilistic models defined through submodular functions, generalizing regular pairwise MRFs and Determinantal Point Processes. In particular, we present LFIELD, a variational approach to general logsubmodular and logsupermodular distributions based on sub and supergradients. We obtain both lower and upper bounds on the logpartition function, which enables us to compute probability intervals for marginals, conditionals and marginal likelihoods. We also obtain fully factorized approximate posteriors, at the same computational cost as ordinary submodular optimization. Our framework results in convex problems for optimizing over differentials of submodular functions, which we show how to optimally solve. We provide theoretical guarantees of the approximation quality with respect to the curvature of the function. We further establish natural relations between our variational approach and the classical meanfield method. Lastly, we empirically demonstrate the accuracy of our inference scheme on several submodular models. 1
Provable submodular minimization using Wolfe’s algorithm
 In NIPS
, 2014
"... Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time [10, 11]. However, these algorithms are typically not practical. In 1976, Wolfe [ ..."
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Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time [10, 11]. However, these algorithms are typically not practical. In 1976, Wolfe [21] proposed an algorithm to find the minimum Euclidean norm point in a polytope, and in 1980, Fujishige [3] showed how Wolfe’s algorithm can be used for SFM. For general submodular functions, this FujishigeWolfe minimum norm algorithm seems to have the best empirical performance. Despite its good practical performance, very little is known about Wolfe’s minimum norm algorithm theoretically. To our knowledge, the only result is an exponential time analysis due to Wolfe [21] himself. In this paper we give a maiden convergence analysis of Wolfe’s algorithm. We prove that in t iterations, Wolfe’s algorithm returns an O(1/t)approximate solution to the minnorm point on any polytope. We also prove a robust version of Fujishige’s theorem which shows that anO(1/n2)approximate solution to the minnorm point on the base polytope implies exact submodular minimization. As a corollary, we get the first pseudopolynomial time guarantee for the FujishigeWolfe minimum norm algorithm for unconstrained submodular function minimization. 1
Submodular Attribute Selection for Action Recognition
 in Video,” NIPS
, 2014
"... In realworld action recognition problems, lowlevel features cannot adequately characterize the rich spatialtemporal structures in action videos. In this work, we encode actions based on attributes that describes actions as highlevel concepts e.g., jump forward or motion in the air. We base our ..."
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In realworld action recognition problems, lowlevel features cannot adequately characterize the rich spatialtemporal structures in action videos. In this work, we encode actions based on attributes that describes actions as highlevel concepts e.g., jump forward or motion in the air. We base our analysis on two types of action attributes. One type of action attributes is generated by humans. The second type is datadriven attributes, which are learned from data using dictionary learning methods. Attributebased representation may exhibit high variance due to noisy and redundant attributes. We propose a discriminative and compact attributebased representation by selecting a subset of discriminative attributes from a large attribute set. Three attribute selection criteria are proposed and formulated as a submodular optimization problem. A greedy optimization algorithm is presented and guaranteed to be at least (11/e)approximation to the optimum. Experimental results on the Olympic Sports and UCF101 datasets demonstrate that the proposed attributebased representation can significantly boost the performance of action recognition algorithms and outperform most recently proposed recognition approaches. 1
Fast Flux Discriminant for LargeScale Sparse Nonlinear Classification
"... In this paper, we propose a novel supervised learning method, Fast Flux Discriminant (FFD), for largescale nonlinear classification. Compared with other existing methods, FFD has unmatched advantages, as it attains the efficiency and interpretability 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 largescale nonlinear classification. Compared with other existing methods, FFD has unmatched advantages, as it attains the efficiency and interpretability 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 density estimation in the entire feature space into selected lowdimensional subspaces. Since there are many possible subspaces, 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 naturally expect nonnegative 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 empirical studies on realworld datasets and show that the proposed model achieves stateoftheart classification results with sparsity, interpretability, and exceptional scalability. Our model can be learned in minutes on datasets with millions of samples, for which most existing nonlinear methods will be prohibitively expensive in space and time.
Streaming Algorithms for Submodular Function Maximization
, 2015
"... We consider the problem of maximizing a nonnegative submodular set function f: 2N → R+ subject to a pmatchoid constraint in the singlepass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result i ..."
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We consider the problem of maximizing a nonnegative submodular set function f: 2N → R+ subject to a pmatchoid constraint in the singlepass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are nonmonotone. We describe deterministic and randomized algorithms that obtain a Ω(1p)approximation using O(k log k)space, where k is an upper bound on the cardinality of the desired set. The model assumes value oracle access to f and membership oracles for the matroids defining the pmatchoid constraint.
On approximate nonsubmodular minimization via treestructured supermodularity.
 In 18th International Conference on Artificial Intelligence and Statistics (AISTATS2015),
, 2015
"... Abstract We address the problem of minimizing nonsubmodular functions where the supermodularity is restricted to treestructured pairwise terms. We are motivated by several real world applications, which require submodularity along with structured supermodularity, and this forms a rich class of ex ..."
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Abstract We address the problem of minimizing nonsubmodular functions where the supermodularity is restricted to treestructured pairwise terms. We are motivated by several real world applications, which require submodularity along with structured supermodularity, and this forms a rich class of expressive models, where the nonsubmodularity is restricted to a tree. While this problem is NP hard (as we show), we develop several practical algorithms to find approximate and nearoptimal solutions for this problem, some of which provide lower and others of which provide upper bounds thereby allowing us to compute a tightness gap for any problem. We compare our algorithms on synthetic data, and also demonstrate the advantage of the formulation on the real world application of image segmentation, where we incorporate structured supermodularity into higherorder submodular energy minimization.
Deep Submodular Functions: Definitions & Learning
"... Abstract We propose and study a new class of submodular functions called deep submodular functions (DSFs). We define DSFs and situate them within the broader context of classes of submodular functions in relationship both to various matroid ranks and sums of concave composed with modular functions ..."
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Abstract We propose and study a new class of submodular functions called deep submodular functions (DSFs). We define DSFs and situate them within the broader context of classes of submodular functions in relationship both to various matroid ranks and sums of concave composed with modular functions (SCMs). Notably, we find that DSFs constitute a strictly broader class than SCMs, thus motivating their use, but that they do not comprise all submodular functions. Interestingly, some DSFs can be seen as special cases of certain deep neural networks (DNNs), hence the name. Finally, we provide a method to learn DSFs in a maxmargin framework, and offer preliminary results applying this both to synthetic and realworld data instances.
On the Reducibility of Submodular Functions
"... Abstract The scalability of submodular optimization methods is critical for their usability in practice. In this paper, we study the reducibility of submodular functions, a property that enables us to reduce the solution space of submodular optimization problems without performance loss. We introdu ..."
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Abstract The scalability of submodular optimization methods is critical for their usability in practice. In this paper, we study the reducibility of submodular functions, a property that enables us to reduce the solution space of submodular optimization problems without performance loss. We introduce the concept of reducibility using marginal gains. Then we show that by adding perturbation, we can endow irreducible functions with reducibility, based on which we propose the perturbationreduction optimization framework. Our theoretical analysis proves that given the perturbation scales, the reducibility gain could be computed, and the performance loss has additive upper bounds. We further conduct empirical studies and the results demonstrate that our proposed framework significantly accelerates existing optimization methods for irreducible submodular functions with a cost of only small performance losses.