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Learning with hypergraphs: Clustering, classification, and embedding
 Advances in Neural Information Processing Systems (NIPS) 19
, 2006
"... We usually endow the investigated objects with pairwise relationships, which can be illustrated as graphs. In many realworld problems, however, relationships among the objects of our interest are more complex than pairwise. Naively squeezing the complex relationships into pairwise ones will inevita ..."
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Cited by 74 (2 self)
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We usually endow the investigated objects with pairwise relationships, which can be illustrated as graphs. In many realworld problems, however, relationships among the objects of our interest are more complex than pairwise. Naively squeezing the complex relationships into pairwise ones will inevitably lead to loss of information which can be expected valuable for our learning tasks however. Therefore we consider using hypergraphs instead to completely represent complex relationships among the objects of our interest, and thus the problem of learning with hypergraphs arises. Our main contribution in this paper is to generalize the powerful methodology of spectral clustering which originally operates on undirected graphs to hypergraphs, and further develop algorithms for hypergraph embedding and transductive classification on the basis of the spectral hypergraph clustering approach. Our experiments on a number of benchmarks showed the advantages of hypergraphs over usual graphs. 1
Higher order learning with graphs
 In ICML ’06: Proceedings of the 23rd international conference on Machine learning
, 2006
"... Recently there has been considerable interest in learning with higher order relations (i.e., threeway or higher) in the unsupervised and semisupervised settings. Hypergraphs and tensors have been proposed as the natural way of representing these relations and their corresponding algebra as the nat ..."
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Cited by 42 (0 self)
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Recently there has been considerable interest in learning with higher order relations (i.e., threeway or higher) in the unsupervised and semisupervised settings. Hypergraphs and tensors have been proposed as the natural way of representing these relations and their corresponding algebra as the natural tools for operating on them. In this paper we argue that hypergraphs are not a natural representation for higher order relations, indeed pairwise as well as higher order relations can be handled using graphs. We show that various formulations of the semisupervised and the unsupervised learning problem on hypergraphs result in the same graph theoretic problem and can be analyzed using existing tools. 1.
Video Object Segmentation by Hypergraph Cut
"... In this paper, we present a new framework of video object segmentation, in which we formulate the task of extracting prominent objects from a scene as the problem of hypergraph cut. We initially oversegment each frame in the sequence, and take the oversegmented image patches as the vertices in the ..."
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Cited by 37 (1 self)
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In this paper, we present a new framework of video object segmentation, in which we formulate the task of extracting prominent objects from a scene as the problem of hypergraph cut. We initially oversegment each frame in the sequence, and take the oversegmented image patches as the vertices in the graph. Different from the traditional pairwise graph structure, we build a novel graph structure, hypergraph, to represent the complex spatiotemporal neighborhood relationship among the patches. We assign each patch with several attributes that are computed from the optical flow and the appearancebased motion profile, and the vertices with the same attribute value is connected by a hyperedge. Through all the hyperedges, not only the complex nonpairwise relationships between the patches are described, but also their merits are integrated together organically. The task of video object segmentation is equivalent to the hypergraph partition, which can be solved by the hypergraph cut algorithm. The effectiveness of the proposed method is demonstrated by extensive experiments on nature scenes. 1.
Foundations of a Multiway Spectral Clustering Framework for Hybrid Linear Modeling
, 2009
"... Abstract The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curva ..."
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Cited by 37 (10 self)
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Abstract The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algorithm for solving the HLM problem and provides careful analysis to justify it. The TSCC algorithm is practically a combination of Govindu’s multiway spectral clustering framework (CVPR 2005) and Ng et al.’s spectral clustering algorithm (NIPS 2001). The main result of this paper states that if the given data is sampled from a mixture of distributions concentrated around affine subspaces, then with high sampling probability the TSCC algorithm segments well the different underlying clusters. The goodness of clustering depends on the withincluster errors, the betweenclusters interaction, and a tuning parameter applied by TSCC. The proof also provides new insights for the analysis of Ng et al. (NIPS 2001). Keywords Hybrid linear modeling · dflats clustering · Multiway clustering · Spectral clustering · Polar curvature · Perturbation analysis · Concentration inequalities Communicated by Albert Cohen. This work was supported by NSF grant #0612608.
A TUTORIAL ON SUBSPACE CLUSTERING
"... The past few years have witnessed an explosion in the availability of data from multiple sources and modalities. For example, millions of cameras have been installed in buildings, streets, airports and cities around the world. This has generated extraordinary advances on how to acquire, compress, st ..."
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Cited by 30 (0 self)
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The past few years have witnessed an explosion in the availability of data from multiple sources and modalities. For example, millions of cameras have been installed in buildings, streets, airports and cities around the world. This has generated extraordinary advances on how to acquire, compress, store, transmit and process massive amounts of complex highdimensional data. Many of these advances have relied on the observation that, even though these data sets are highdimensional, their intrinsic dimension is often much smaller than the dimension of the ambient space. In computer vision, for example, the number of pixels in an image can be rather large, yet most computer vision models use only a few parameters to describe the appearance, geometry and dynamics of a scene. This has motivated the development of a number of techniques for finding a lowdimensional representation
A GameTheoretic Approach to Hypergraph Clustering
, 2009
"... Hypergraph clustering refers to the process of extracting maximally coherent groups from a set of objects using highorder (rather than pairwise) similarities. Traditional approaches to this problem are based on the idea of partitioning the input data into a userdefined number of classes, thereby o ..."
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Cited by 26 (2 self)
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Hypergraph clustering refers to the process of extracting maximally coherent groups from a set of objects using highorder (rather than pairwise) similarities. Traditional approaches to this problem are based on the idea of partitioning the input data into a userdefined number of classes, thereby obtaining the clusters as a byproduct of the partitioning process. In this paper, we provide a radically different perspective to the problem. In contrast to the classical approach, we attempt to provide a meaningful formalization of the very notion of a cluster and we show that game theory offers an attractive and unexplored perspective that serves well our purpose. Specifically, we show that the hypergraph clustering problem can be naturally cast into a noncooperative multiplayer “clustering game”, whereby the notion of a cluster is equivalent to a classical gametheoretic equilibrium concept. From the computational viewpoint, we show that the problem of finding the equilibria of our clustering game is equivalent to locally optimizing a polynomial function over the standard simplex, and we provide a discretetime dynamics to perform this optimization. Experiments are presented which show the superiority of our approach over stateoftheart hypergraph clustering techniques.
Higher order motion models and spectral clustering
 In CVPR
, 2012
"... Motion segmentation based on point trajectories can integrate information of a whole video shot to detect and separate moving objects. Commonly, similarities are defined between pairs of trajectories. However, pairwise similarities restrict the motion model to translations. Nontranslational motion, ..."
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Cited by 25 (2 self)
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Motion segmentation based on point trajectories can integrate information of a whole video shot to detect and separate moving objects. Commonly, similarities are defined between pairs of trajectories. However, pairwise similarities restrict the motion model to translations. Nontranslational motion, such as rotation or scaling, is penalized in such an approach. We propose to define similarities on higher order tuples rather than pairs, which leads to hypergraphs. To apply spectral clustering, the hypergraph is transferred to an ordinary graph, an operation that can be interpreted as a projection. We propose a specific nonlinear projection via a regularized maximum operator, and show that it yields significant improvements both compared to pairwise similarities and alternative hypergraph projections. 1.
Image Retrieval via Probabilistic Hypergraph Ranking
"... In this paper, we propose a new transductive learning framework for image retrieval, in which images are taken as vertices in a weighted hypergraph and the task of image search is formulated as the problem of hypergraph ranking. Based on the similarity matrix computed from various feature descriptor ..."
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Cited by 24 (0 self)
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In this paper, we propose a new transductive learning framework for image retrieval, in which images are taken as vertices in a weighted hypergraph and the task of image search is formulated as the problem of hypergraph ranking. Based on the similarity matrix computed from various feature descriptors, we take each image as a ‘centroid ’ vertex and form a hyperedge by a centroid and its knearest neighbors. To further exploit the correlation information among images, we propose a probabilistic hypergraph, which assigns each vertex vi to a hyperedge ej in a probabilistic way. In the incidence structure of a probabilistic hypergraph, we describe both the higher order grouping information and the affinity relationship between vertices within each hyperedge. After feedback images are provided, our retrieval system ranks image labels by a transductive inference approach, which tends to assign the same label to vertices that share many incidental hyperedges, with the constraints that predicted labels of feedback images should be similar to their initial labels. We compare the proposed method to several other methods and its effectiveness is demonstrated by extensive experiments on Corel5K, the Scene dataset and Caltech 101. 1.
Normalized tree partitionning for image segmentation
 in Proc. IEEE Int. Conf. Comput. Vis. Pattern Recog
, 2008
"... In this paper, we propose a novel graph based clustering approach with satisfactory clustering performance and low computational cost. It consists of two main steps: tree fitting and partitioning. We first introduce a probabilistic method to fit a tree to a data graph under the sense of minimum entr ..."
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Cited by 20 (3 self)
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In this paper, we propose a novel graph based clustering approach with satisfactory clustering performance and low computational cost. It consists of two main steps: tree fitting and partitioning. We first introduce a probabilistic method to fit a tree to a data graph under the sense of minimum entropy. Then, we propose a novel tree partitioning method under a normalized cut criterion, called Normalized Tree Partitioning (NTP), in which a fast combinatorial algorithm is designed for exact bipartitioning. Moreover, we extend it to kway tree partitioning by proposing an efficient bestfirst recursive bipartitioning scheme. Compared with spectral clustering, NTP produces the exact global optimal bipartition, introduces fewer approximations for kway partitioning and can intrinsically produce superior performance. Compared with bottomup aggregation methods, NTP adopts a global criterion and hence performs better. Last, experimental results on image segmentation demonstrate that our approach is more powerful compared with existing graphbased approaches. 1.
Spectral clustering based on local linear approximations
 ELECTRONIC JOURNAL OF STATISTICS
, 2011
"... Abstract: In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points sampled in space away from the clusters. We consider a ..."
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Cited by 18 (5 self)
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Abstract: In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points sampled in space away from the clusters. We consider a prototype for a higherorder spectral clustering method based on the residual from a local linear approximation. We obtain theoretical guarantees for this algorithm and show that, in terms of both separation and robustness to outliers, it outperforms the standard spectral clustering algorithm (based on pairwise distances) of Ng, Jordan and Weiss (NIPS ’01). The optimal choice for some of the tuning parameters depends on the dimension and thickness of the clusters. We provide estimators that come close enough for our theoretical purposes. We also discuss the cases of clusters of mixed dimensions and of clusters that are generated from smoother surfaces. In our experiments, this algorithm is shown to outperform pairwise spectral clustering on both simulated and real data.