Results 1  10
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22
The spectral norm error of the naive nystrom extension
, 2011
"... Abstract. The näıve Nyström extension forms a lowrank approximation to a positivesemidefinite matrix by uniformly randomly sampling from its columns. This paper provides the first relativeerror bound on the spectral norm error incurred in this process. This bound follows from a natural connecti ..."
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Abstract. The näıve Nyström extension forms a lowrank approximation to a positivesemidefinite matrix by uniformly randomly sampling from its columns. This paper provides the first relativeerror bound on the spectral norm error incurred in this process. This bound follows from a natural connection between the Nyström extension and the column subset selection problem. The main tool is a matrix Chernoff bound for sampling without replacement. 1.
An MBO scheme on graphs for segmentation and image processing
, 2012
"... In this paper we present a computationally efficient algorithm utilizing a fully or semi nonlocal graph Laplacian for solving a wide range of learning problems in data clustering and image processing. Combining ideas from L1 compressive sensing, image processing and graph methods, the diffuse inte ..."
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Cited by 10 (5 self)
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In this paper we present a computationally efficient algorithm utilizing a fully or semi nonlocal graph Laplacian for solving a wide range of learning problems in data clustering and image processing. Combining ideas from L1 compressive sensing, image processing and graph methods, the diffuse interface model based on the GinzburgLandau functional was recently introduced to the graph community for solving problems in data classification. Here, we propose an adaptation of the classic numerical MerrimanBenceOsher (MBO) scheme for graphbased methods and also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian. We present various computational examples to demonstrate the performance of our model, which is successful on images with texture and repetitive structure due to its nonlocal nature.
Multiclass total variation clustering
 In Advances in Neural Information Processing Systems (NIPS
, 2013
"... Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bipartitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. Thi ..."
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Cited by 8 (2 self)
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Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bipartitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a general framework for multiclass total variation clustering that does not rely on recursion. The results greatly outperform previous total variation algorithms and compare well with stateoftheart NMF approaches. 1
Fast Multiclass Segmentation using Diffuse Interface Methods on Graphs
"... We present two graphbased algorithms for multiclass segmentation of highdimensional data. The algorithms use a diffuse interface model based on the GinzburgLandau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs ..."
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Cited by 5 (2 self)
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We present two graphbased algorithms for multiclass segmentation of highdimensional data. The algorithms use a diffuse interface model based on the GinzburgLandau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional’s doublewell potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical MerrimanBenceOsher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current stateoftheart multiclass segmentation algorithms.
An MBO scheme on graphs for classification and image processing
 SIAM J. IMAGING SCI
, 2013
"... In this paper we present a computationally efficient algorithm utilizing a fully or seminonlocal graph Laplacian for solving a wide range of learning problems in binary data classification and image processing. In their recent work [Multiscale Model. Simul., 10 (2012), pp. 1090–1118], Bertozzi and ..."
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Cited by 5 (2 self)
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In this paper we present a computationally efficient algorithm utilizing a fully or seminonlocal graph Laplacian for solving a wide range of learning problems in binary data classification and image processing. In their recent work [Multiscale Model. Simul., 10 (2012), pp. 1090–1118], Bertozzi and Flenner introduced a graphbased diffuse interface model utilizing the Ginzburg–Landau functional for solving problems in data classification. Here, we propose an adaptation of the classic numerical Merriman–Bence–Osher (MBO) scheme for minimizing graphbased diffuse interface functionals, like those originally proposed by Bertozzi and Flenner. We also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian. Various computational examples are presented to demonstrate the performance of our algorithm, which is successful on images with texture and repetitive structure due to its nonlocal nature. The results show that our method is multiple times more efficient than other wellknown nonlocal models.
Asymmetric Cheeger cut and application to multiclass unsupervised clustering
, 2012
"... Cheeger cut has recently been shown to provide excellent classification results for two classes. Whereas the classical Cheeger cut favors a 5050 partition of the graph, we present here an asymmetric variant of the Cheeger cut which favors, for example, a 1090 partition. This asymmetric Cheeger cut ..."
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Cheeger cut has recently been shown to provide excellent classification results for two classes. Whereas the classical Cheeger cut favors a 5050 partition of the graph, we present here an asymmetric variant of the Cheeger cut which favors, for example, a 1090 partition. This asymmetric Cheeger cut provides a powerful tool for unsupervised multiclass partitioning. We use it in recursive bipartitioning to detach one after the other each of the classes. This asymmetric recursive algorithm handles equally well any number of classes, as opposed to symmetric recursive bipartitioning which is naturally better suited for 2m classes. We obtain an error classification rate of 2.35 % and 4.07 % for MNIST and USPS benchmark datasets respectively, drastically improving the former 11.7 % and 13 % error rate obtained in the literature with symmetric Cheeger cut bipartitioning algorithms. 1
A METHOD BASED ON TOTAL VARIATION FOR NETWORK MODULARITY OPTIMIZATION USING THE MBO SCHEME∗
"... Abstract. The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called “communities. ” One popular approach to finding communities i ..."
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Cited by 2 (1 self)
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Abstract. The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called “communities. ” One popular approach to finding communities is to maximize a quality function known as modularity to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an 2 balance term. By employing numerical techniques from image processing and 1 compressive sensing—such as convex splitting and the Merriman–Bence–Osher (MBO) scheme—we develop a variational algorithm for the minimization problem. We present our computational results using both synthetic benchmark networks and real data.
CONTINUUM LIMIT OF TOTAL VARIATION ON POINT CLOUDS
, 2014
"... We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, a ..."
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Cited by 2 (0 self)
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We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graphbased machine learning algorithms for tasks such as clustering. In particular, we study when is the cut capacity, and more generally total variation, on these graphs a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γconvergence. We obtain almost optimal conditions on the scaling, as number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γconvergence to hold. Taking the limit is enabled by a new metric which allows to suitably compare functionals defined on different point clouds.
Sparse representation on graphs by tight wavelet frames and applications
, 2014
"... Abstract. In this paper, we introduce a unified theory of tight wavelet frames on nonflat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph ..."
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Abstract. In this paper, we introduce a unified theory of tight wavelet frames on nonflat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start from defining multiresolution analysis (MRA) generated by a single generator on manifolds, and discuss the conditions needed for a generator to form an MRA. With a given MRA, we show how MRAbased tight frames can be constructed and the conditions needed for them to generate a tight wavelet frame on manifolds. In particular, we show that under suitable conditions, framelets constructed from the unitary extension principle [1] can also generate tight frame systems on manifolds. We also discuss how the transition from the continuum to the discrete setting can be naturally defined, which leads to the multilevel discrete tight wavelet frame transforms (decomposition and reconstruction) on graphs. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately. More importantly, numerical simulations show that the proposed discrete tight wavelet frame transform maps piecewise smooth data to a set of sparse coefficients. This indicates that the proposed tight wavelet frames indeed provide sparse representation on graphs. Finally, we consider two specific applications: graph data denoising and semisupervised clustering. Utilizing the proposed sparse representation, we introduce `1norm based optimization models for denoising and semisupervised clustering, which are inspired by the models used in image restoration and image segmentation. 1.
Multiclass diffuse interface models for semisupervised learning on graphs
 in Proceedings of the 2th International Conference on Pattern Recognition Applications and Methods. SciTePress
, 2013
"... Abstract: We present a graphbased variational algorithm for multiclass classification of highdimensional data, motivated by total variation techniques. The energy functional is based on a diffuse interface model with a periodic potential. We augment the model by introducing an alternative measure ..."
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Abstract: We present a graphbased variational algorithm for multiclass classification of highdimensional data, motivated by total variation techniques. The energy functional is based on a diffuse interface model with a periodic potential. We augment the model by introducing an alternative measure of smoothness that preserves symmetry among the class labels. Through this modification of the standard Laplacian, we construct an efficient multiclass method that allows for sharp transitions between classes. The experimental results demonstrate that our approach is competitive with the state of the art among other graphbased algorithms. 1