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4,456
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
Abstract
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Cited by 819 (28 self)
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of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing
The Quickhull algorithm for convex hulls
- ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algo ..."
Abstract
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Cited by 713 (0 self)
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is implemented with floating-point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation
Loopy belief propagation for approximate inference: An empirical study. In:
- Proceedings of Uncertainty in AI,
, 1999
"... Abstract Recently, researchers have demonstrated that "loopy belief propagation" -the use of Pearl's polytree algorithm in a Bayesian network with loops -can perform well in the context of error-correcting codes. The most dramatic instance of this is the near Shannon-limit performanc ..."
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Cited by 676 (15 self)
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in a more gen eral setting? We compare the marginals com puted using loopy propagation to the exact ones in four Bayesian network architectures, including two real-world networks: ALARM and QMR. We find that the loopy beliefs of ten converge and when they do, they give a good approximation
Clustering with Bregman Divergences
- JOURNAL OF MACHINE LEARNING RESEARCH
, 2005
"... A wide variety of distortion functions are used for clustering, e.g., squared Euclidean distance, Mahalanobis distance and relative entropy. In this paper, we propose and analyze parametric hard and soft clustering algorithms based on a large class of distortion functions known as Bregman divergence ..."
Abstract
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Cited by 443 (57 self)
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generalizing the basic idea to a very large class of clustering loss functions. There are two main contributions in this paper. First, we pose the hard clustering problem in terms of minimizing the loss in Bregman information, a quantity motivated by rate-distortion theory, and present an algorithm to minimize
The slider-pinning problem
- CCCG
, 2007
"... A Laman mechanism is a flexible planar bar-and-joint framework with m ≤ 2n − 3 edges and exactly k = 2n − m degrees of freedom. The slider-pinning problem is to eliminate all the degrees of freedom of a Laman mechanism, in an optimal fashion, by individually fixing x or y coordinates of vertices. We ..."
Abstract
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Cited by 384 (7 self)
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A Laman mechanism is a flexible planar bar-and-joint framework with m ≤ 2n − 3 edges and exactly k = 2n − m degrees of freedom. The slider-pinning problem is to eliminate all the degrees of freedom of a Laman mechanism, in an optimal fashion, by individually fixing x or y coordinates of vertices
The Complexity of Decentralized Control of Markov Decision Processes
- Mathematics of Operations Research
, 2000
"... We consider decentralized control of Markov decision processes and give complexity bounds on the worst-case running time for algorithms that find optimal solutions. Generalizations of both the fullyobservable case and the partially-observable case that allow for decentralized control are described. ..."
Abstract
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Cited by 411 (46 self)
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. For even two agents, the finite-horizon problems corresponding to both of these models are hard for nondeterministic exponential time. These complexity results illustrate a fundamental difference between centralized and decentralized control of Markov decision processes. In contrast to the problems
Complexity of finding embeddings in a k-tree
- SIAM JOURNAL OF DISCRETE MATHEMATICS
, 1987
"... A k-tree is a graph that can be reduced to the k-complete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a k-tree. This problem is motivated by the existence of polynomial time al ..."
Abstract
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Cited by 386 (1 self)
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status of two problems related to finding the smallest number k such that a given graph is a partial k-tree. First, the corresponding decision problem is NP-complete. Second, for a fixed (predetermined) value of k, we present an algorithm with polynomially bounded (but exponential in k) worst case time
The Web as a graph: measurements, models, and methods
, 1999
"... . The pages and hyperlinks of the World-Wide Web may be viewed as nodes and edges in a directed graph. This graph is a fascinating object of study: it has several hundred million nodes today, over a billion links, and appears to grow exponentially with time. There are many reasons --- mathematical, ..."
Abstract
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Cited by 373 (11 self)
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, sociological, and commercial --- for studying the evolution of this graph. In this paper we begin by describing two algorithms that operate on the Web graph, addressing problems from Web search and automatic community discovery. We then report a number of measurements and properties of this graph
A Fast Algorithm for the Minimum Covariance Determinant Estimator
- Technometrics
, 1998
"... The minimum covariance determinant (MCD) method of Rousseeuw (1984) is a highly robust estimator of multivariate location and scatter. Its objective is to find h observations (out of n) whose covariance matrix has the lowest determinant. Until now applications of the MCD were hampered by the comput ..."
Abstract
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Cited by 346 (15 self)
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by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size: one about a production process at Philips with n = 677 objects and p = 9 variables, and a data set from astronomy with n =137,256 objects and p = 27
Efficient algorithms for processing XPath queries
- In VLDB
, 2002
"... Our experimental analysis of several popular XPath processors reveals a striking fact: Query evaluation in each of the systems requires time exponential in the size of queries in the worst case. We show that XPath can be processed much more efficiently, and propose main-memory algorithms for this pr ..."
Abstract
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Cited by 306 (23 self)
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for this problem with polynomial-time combined query evaluation complexity. Moreover, we present two fragments of XPath for which linear-time query processing algorithms exist. 1
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