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69
Atomic decomposition by basis pursuit
 SIAM Journal on Scientific Computing
, 1998
"... Abstract. The timefrequency and timescale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several meth ..."
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Cited by 1673 (43 self)
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Abstract. The timefrequency and timescale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an “optimal ” superposition of dictionary elements, where optimal means having the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as illposed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to largescale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interiorpoint methods. We obtain reasonable success with a primaldual logarithmic barrier method and conjugategradient solver.
Outlier Detection and Motion Segmentation
, 1995
"... this paper we examine methods for the detection of outliers to a least squares fit that would have been previously computationally infeasible. The fitting of linear regression models by least squares is undoubtedly the most widely used modelling procedure. A major drawback, however, is that outliers ..."
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Cited by 101 (16 self)
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this paper we examine methods for the detection of outliers to a least squares fit that would have been previously computationally infeasible. The fitting of linear regression models by least squares is undoubtedly the most widely used modelling procedure. A major drawback, however, is that outliers which are inevitably included in the initial fit can so distort the fitting process that the resulting fit can be arbitrary. A common practice is to search for outliers using the raw residuals. However, the use of these on their own can be misleading. Much work has been done already on detecting outliers given the case of nonorthogonal regression (reviewed in [2]), where we choose to regress against a given, dependent, variable. Unfortunately there is in this method a tacit assumption that all the error is concentrated in the dependent variable and furthermore that the dependent variable has a nonzero coefficient. In many engineering situations we cannot guarantee these conditions and we must resort to orthogonal regression, where we minimize the sum of squares of the perpendicular distances (i.e. the residuals) between each point and the fitted hyperplane. Little work has been done on outlier detection for orthogonal regression, with the exception of [15]. In this paper we outline two methodologies for outlier detection. In sections 25 we describe an extension of previous outlier diagnostics to the realm of orthogonal regression. The method works by assessing the amount of influence that the deletion of each point would have on the final solution. In sections 67 we then apply the theory we have developed to the calculation of the Fundamental matrixa necessary first step in many structure and motion algorithms. Finally in section 8 we outline an alternative approach...
Hessian Eigenmaps: New Locally Linear Embedding Techniques For HighDimensional Data
, 2003
"... We describe a method to recover the underlying parametrization of scattered data (m i ) lying on a manifold M embedded in highdimensional Euclidean space. The method, Hessianbased Locally Linear Embedding (HLLE), derives from a conceptual framework of Local Isometry in which the manifold M , viewe ..."
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Cited by 99 (0 self)
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We describe a method to recover the underlying parametrization of scattered data (m i ) lying on a manifold M embedded in highdimensional Euclidean space. The method, Hessianbased Locally Linear Embedding (HLLE), derives from a conceptual framework of Local Isometry in which the manifold M , viewed as a Riemannian submanifold of the ambient Euclidean space R , is locally isometric to an open, connected subset # of Euclidean space R . Since # does not have to be convex, this framework is able to handle a significantly wider class of situations than the original Isomap algorithm.
CubeSVD: A Novel Approach to Personalized Web Search
 In Proc. of the 14 th International World Wide Web Conference (WWW
, 2005
"... As the competition of Web search market increases, there is a high demand for personalized Web search to conduct retrieval incorporating Web users' information needs. This paper focuses on utilizing clickthrough data to improve Web search. Since millions of searches are conducted everyday, a se ..."
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Cited by 78 (3 self)
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As the competition of Web search market increases, there is a high demand for personalized Web search to conduct retrieval incorporating Web users' information needs. This paper focuses on utilizing clickthrough data to improve Web search. Since millions of searches are conducted everyday, a search engine accumulates a large volume of clickthrough data, which records who submits queries and which pages he/she clicks on. The clickthrough data is highly sparse and contains di#erent types of objects (user, query and Web page), and the relationships among these objects are also very complicated. By performing analysis on these data, we attempt to discover Web users' interests and the patterns that users locate information. In this paper, a novel approach CubeSVD is proposed to improve Web search. The clickthrough data is represented by a 3order tensor, on which we perform 3mode analysis using the higherorder singular value decomposition technique to automatically capture the latent factors that govern the relations among these multitype objects: users, queries and Web pages. A tensor reconstructed based on the CubeSVD analysis reflects both the observed interactions among these objects and the implicit associations among them. Therefore, Web search activities can be carried out based on CubeSVD analysis. Experimental evaluations using a realworld data set collected from an MSN search engine show that CubeSVD achieves encouraging search results in comparison with some standard methods.
Blocking and Array Contraction Across Arbitrarily Nested Loops Using Affine Partitioning
, 2001
"... Applicable to arbitrary sequences and nests of loops, affine partitioning is a program transformation framework that unifies many previously proposed loop transformations, including unimodular transforms, fusion, fission, reindexing, scaling and statement reordering. Algorithms based on affine parti ..."
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Cited by 67 (1 self)
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Applicable to arbitrary sequences and nests of loops, affine partitioning is a program transformation framework that unifies many previously proposed loop transformations, including unimodular transforms, fusion, fission, reindexing, scaling and statement reordering. Algorithms based on affine partitioning have been shown to be effective for parallelization and communication minimization. This paper presents algorithms that improve data locality using affine partitioning. Blocking and array contraction are two important optimizations that have been shown to be useful for data locality. Blocking creates a set of inner loops so that data brought into the faster levels of the memory hierarchy can be reused. Array contraction reduces an array to a scalar variable and thereby reduces the number of memory operations executed and the memory footprint. Loop transforms are often necessary to make blocking and array contraction possible.
A Cost Model for Query Processing in HighDimensional Data Spaces
, 2000
"... During the last decade, multimedia databases have become increasingly important in many application areas such as medicine, CAD, geography or molecular biology. An important research issue in the field of multimedia databases is similarity search in large data sets. Most current approaches addressin ..."
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Cited by 47 (0 self)
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During the last decade, multimedia databases have become increasingly important in many application areas such as medicine, CAD, geography or molecular biology. An important research issue in the field of multimedia databases is similarity search in large data sets. Most current approaches addressing similarity search use the socalled feature approach which transforms important properties of the stored objects into points of a highdimensional space (feature vectors). Thus, the similarity search is transformed into a neighborhood search in the feature space. For the management of the feature vectors, multidimensional index structures are usually applied. The performance of query processing can be substantially improved by opti...
Concept Hierarchy Based Text Database Categorization
, 2000
"... Document categorization as a technique to improve the retrieval of useful documents has been extensively investigated. One important issue in a largescale metasearch engine is to select text databases that are likely to contain useful documents for a given query. We believe that database categoriza ..."
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Cited by 37 (6 self)
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Document categorization as a technique to improve the retrieval of useful documents has been extensively investigated. One important issue in a largescale metasearch engine is to select text databases that are likely to contain useful documents for a given query. We believe that database categorization can be a potentially effective technique for good database selection, especially in the Internet environment where short queries are usually submitted. In this paper, we propose and evaluate several database categorization algorithms. This study indicates that while some document categorization algorithms could be adopted for database categorization, algorithms that take into consideration the special characteristics of databases may be more effective. Preliminary experimental results are provided to compare the proposed database categorization algorithms. A prototype database categorization system based on one of the proposed algorithms has been developed.
Approximate Inverse Preconditioners for General Sparse Matrices
, 1994
"... The standard Incomplete LU (ILU) preconditioners often fail for general sparse indefinite matrices because they give rise to `unstable' factors L and U . In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse precondit ..."
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Cited by 25 (6 self)
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The standard Incomplete LU (ILU) preconditioners often fail for general sparse indefinite matrices because they give rise to `unstable' factors L and U . In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse preconditioners based on minimizing kI \GammaAM k F , where AM is the preconditioned matrix. An iterative descenttype method is used to approximate each column of the inverse. For this approach to be efficient, the iteration must be done in sparse mode, i.e., with `sparsematrix by sparsevector' operations. Numerical dropping is applied to each column to maintain sparsity in the approximate inverse. Compared to previous methods, this is a natural way to determine the sparsity pattern of the approximate inverse. This paper discusses options such as Newton and `global' iteration, selfpreconditioning, dropping strategies, and factorized forms. The performance of the options are compared on standar...
Transductive classification via local learning regularization. AISTATS
, 2007
"... The idea of local learning, classifying a particular point based on its neighbors, has been successfully applied to supervised learning problems. In this paper, we adapt it for Transductive Classification (TC) problems. Specifically, we formulate a Local Learning Regularizer (LLReg) which leads to ..."
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Cited by 18 (0 self)
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The idea of local learning, classifying a particular point based on its neighbors, has been successfully applied to supervised learning problems. In this paper, we adapt it for Transductive Classification (TC) problems. Specifically, we formulate a Local Learning Regularizer (LLReg) which leads to a solution with the property that the label of each data point can be well predicted based on its neighbors and their labels. For model selection, an efficient way to compute the leaveoneout classification error is provided for the proposed and related algorithms. Experimental results using several benchmark datasets illustrate the effectiveness of the proposed approach. 1