Results 1  10
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12
Modelbased overlapping clustering
 In KDD
, 2005
"... While the vast majority of clustering algorithms are partitional, many real world datasets have inherently overlapping clusters. Several approaches to finding overlapping clusters have come from work on analysis of biological datasets. In this paper, we interpret an overlapping clustering model prop ..."
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Cited by 29 (6 self)
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While the vast majority of clustering algorithms are partitional, many real world datasets have inherently overlapping clusters. Several approaches to finding overlapping clusters have come from work on analysis of biological datasets. In this paper, we interpret an overlapping clustering model proposed by Segal et al. [23] as a generalization of Gaussian mixture models, and we extend it to an overlapping clustering model based on mixtures of any regular exponential family distribution and the corresponding Bregman divergence. We provide the necessary algorithm modifications for this extension, and present results on synthetic data as well as subsets of 20Newsgroups and EachMovie datasets.
Overview of methods for image reconstruction from projections in emission computed tomography
 PROC. IEEE
, 2003
"... Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in t ..."
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Cited by 18 (1 self)
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Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in the instrumentation for data collection, and in the computer methods for generating images from the measured data. These computer methods are designed to solve the inverse problem known as “image reconstruction from projections.” This paper uses the various models of the data collection process as the framework for presenting an overview of the wide variety of methods that have been developed for image reconstruction in the major subfields of ECT, which are positron emission tomography (PET) and singlephoton emission computed tomography (SPECT). The overall sequence of the major sections in the paper, and the presentation within each major section, both proceed from the more realistic and general models to those that are idealized and application specific. For most of the topics, the description proceeds from the threedimensional case to the twodimensional case. The paper presents a broad overview of algorithms for PET and SPECT, giving references to the literature where these algorithms and their applications are described in more detail.
Alternating minimization and projection methods for nonconvex problems
 0801.1780v2[math.oc], arXiv
, 2008
"... Abstract We study the convergence properties of alternating proximal minimization algorithms for (nonconvex) functions of the following type: L(x,y) = f(x) + Q(x,y) + g(y) where f: R n → R∪{+∞} and g: R m → R∪{+∞} are proper lower semicontinuous functions and Q: R n ×R m → R is a smooth C 1 (finite ..."
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Cited by 15 (2 self)
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Abstract We study the convergence properties of alternating proximal minimization algorithms for (nonconvex) functions of the following type: L(x,y) = f(x) + Q(x,y) + g(y) where f: R n → R∪{+∞} and g: R m → R∪{+∞} are proper lower semicontinuous functions and Q: R n ×R m → R is a smooth C 1 (finite valued) function which couples the variables x and y. The algorithm is defined by: (x0, y0) ∈ R n × R m given, (xk, yk) → (xk+1, yk) → (xk+1, yk+1)
Joint minimization with alternating Bregman proximity operators
 Pac. J. Optim
, 2005
"... Abstract A systematic study of the proximity properties of Bregman distances is carried out. Thisinvestigation leads to the introduction of a new type of proximity operator which complements the usual Bregman proximity operator. We establish key properties of these operators andutilize them to devis ..."
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Cited by 8 (2 self)
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Abstract A systematic study of the proximity properties of Bregman distances is carried out. Thisinvestigation leads to the introduction of a new type of proximity operator which complements the usual Bregman proximity operator. We establish key properties of these operators andutilize them to devise a new alternating procedure for solving a broad class of joint minimization problems. We provide a comprehensive convergence analysis of this algorithm. Our frameworkis shown to capture and extend various optimization methods.
Iterating Bregman Retractions
, 2002
"... The notion of a Bregman retraction of a closed convex set in Euclidean space is introduced. Bregman retractions include backward Bregman projections, forward Bregman projections, as well as their convex combinations, and are thus quite exible. The main result on iterating Bregman retractions unifies ..."
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Cited by 3 (2 self)
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The notion of a Bregman retraction of a closed convex set in Euclidean space is introduced. Bregman retractions include backward Bregman projections, forward Bregman projections, as well as their convex combinations, and are thus quite exible. The main result on iterating Bregman retractions unifies several convergence results on projection methods for solving convex feasibility problems. It is also used to construct new sequential and parallel algorithms.
Alternating minimization algorithms for transmission tomography
 IEEE Trans. Med. Imag
, 2001
"... A family of alternating minimization algorithms for finding maximum likelihood estimates of attenuation functions in transmission xray tomography is described. The model from which the algorithms are derived includes polyenergetic photon spectra, background events, and nonideal point spread functio ..."
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Cited by 3 (0 self)
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A family of alternating minimization algorithms for finding maximum likelihood estimates of attenuation functions in transmission xray tomography is described. The model from which the algorithms are derived includes polyenergetic photon spectra, background events, and nonideal point spread functions. The maximum likelihood image reconstruction problem is reformulated as a double minimization of the Idivergence. A novel application of the convex decomposition lemma results in an alternating minimization algorithm that monotonically decreases the objective function. Each step of the minimization is in closed form. The family of algorithms includes variations that use ordered subset techniques for increasing the speed of convergence. Simulations demonstrate the ability to correct the cupping artifact due to beam hardening and the ability to reduce streaking artifacts that arise from beam hardening and background events.
ON DIAGONALLYRELAXED ORTHOGONAL PROJECTION METHODS
, 2005
"... Abstract. We propose and study a blockiterative projections method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonallyrelaxed or ..."
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Cited by 2 (1 self)
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Abstract. We propose and study a blockiterative projections method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonallyrelaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and blockiterative projection algorithms but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.
A Proximal Point Algorithm with Bregman Distances for Quasiconvex Optimization over the Positive Orthant
"... We present an interior proximal point method with Bregman distance, whose Bregman function is separable and the zone is the interior of the positive orthant, for solving the quasiconvex optimization problem under nonnegative constraints. We establish that the sequence generated by our algorithm is w ..."
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Cited by 1 (1 self)
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We present an interior proximal point method with Bregman distance, whose Bregman function is separable and the zone is the interior of the positive orthant, for solving the quasiconvex optimization problem under nonnegative constraints. We establish that the sequence generated by our algorithm is well defined and we prove convergence to a solution point when the sequence of parameters goes to zero. When the parameters are bounded above, we get the convergence to a KKT point.
The Linearized Bregman Method via Split Feasibility Problems: Analysis and Generalizations ∗
"... Abstract. The linearized Bregman method is a method to calculate sparse solutions to systems of linear equations. We formulate this problem as a split feasibility problem, propose an algorithmic framework based on Bregman projections, and prove a general convergence result for this framework. Conver ..."
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Cited by 1 (1 self)
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Abstract. The linearized Bregman method is a method to calculate sparse solutions to systems of linear equations. We formulate this problem as a split feasibility problem, propose an algorithmic framework based on Bregman projections, and prove a general convergence result for this framework. Convergence of the linearized Bregman method will be obtained as a special case. Our approach also allows for several generalizations such as other objective functions, incremental iterations, incorporation of nonGaussian noise models, and box constraints.
AN INCOMPLETE PROJECTIONS ALGORITHM FOR SOLVING LARGE INCONSISTENT LINEAR SYSTEMS
"... The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate x k+1 by projecting the current point x k onto a separating hyperplane generated by a given linear combination of the original hyperplanes and/or halfspaces. The authors have ..."
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The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate x k+1 by projecting the current point x k onto a separating hyperplane generated by a given linear combination of the original hyperplanes and/or halfspaces. The authors have introduced in several papers new acceleration schemes for solving systems of linear equations and inequalities respectively, within a PAM like framework. The basic idea was to force the next iterate to belong to the convex region defined by the new separating or aggregated hyperplane computed in the previous iteration. In this paper the above mentioned methods are extended to the problem of finding the least squares solution to inconsistent systems. The new algorithm is based upon a new scheme of incomplete alternate projections for minimizing the proximity function. The parallel simultaneous projections ACCIM algorithm, published by the authors, is the basis for calculating the incomplete intermediate projections. The convergence properties of the new algorithm are given together with numerical experiences obtained by applying it to image reconstruction problems using the SNARK93 system.