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Convolution Kernels on Discrete Structures

by David Haussler , 1999
"... We introduce a new method of constructing kernels on sets whose elements are discrete structures like strings, trees and graphs. The method can be applied iteratively to build a kernel on an infinite set from kernels involving generators of the set. The family of kernels generated generalizes the fa ..."
Abstract - Cited by 506 (0 self) - Add to MetaCart
the family of radial basis kernels. It can also be used to define kernels in the form of joint Gibbs probability distributions. Kernels can be built from hidden Markov random elds, generalized regular expressions, pair-HMMs, or ANOVA decompositions. Uses of the method lead to open problems involving

Learning the Kernel Matrix with Semi-Definite Programming

by Gert R. G. Lanckriet, Nello Cristianini, Laurent El Ghaoui, Peter Bartlett, Michael I. Jordan , 2002
"... Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information ..."
Abstract - Cited by 775 (21 self) - Add to MetaCart
is contained in the so-called kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space---classical model selection

Estimating the Support of a High-Dimensional Distribution

by Bernhard Schölkopf, John C. Platt, John Shawe-taylor, Alex J. Smola, Robert C. Williamson , 1999
"... Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propo ..."
Abstract - Cited by 783 (29 self) - Add to MetaCart
propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length

Iterative point matching for registration of free-form curves and surfaces

by Zhengyou Zhang , 1994
"... A heuristic method has been developed for registering two sets of 3-D curves obtained by using an edge-based stereo system, or two dense 3-D maps obtained by using a correlation-based stereo system. Geometric matching in general is a difficult unsolved problem in computer vision. Fortunately, in ma ..."
Abstract - Cited by 660 (8 self) - Add to MetaCart
A heuristic method has been developed for registering two sets of 3-D curves obtained by using an edge-based stereo system, or two dense 3-D maps obtained by using a correlation-based stereo system. Geometric matching in general is a difficult unsolved problem in computer vision. Fortunately

Loopy belief propagation for approximate inference: An empirical study. In:

by Kevin P Murphy , Yair Weiss , Michael I Jordan - 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 ..."
Abstract - Cited by 676 (15 self) - Add to MetaCart
tiply connected networks: When loops are present, the network is no longer singly connected and local propaga tion schemes will invariably run into trouble . We believe there are general undiscovered theorems about the performance of belief propagation on loopy DAGs. These theo rems, which may have

Combinatorial Geometry

by P. K. Agarwal , 1995
"... Abstract. Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following F-range searching problem: Given P, build a data stru ..."
Abstract - Cited by 185 (24 self) - Add to MetaCart
Abstract. Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following F-range searching problem: Given P, build a data

Graph embedding and extension: A general framework for dimensionality reduction

by Shuicheng Yan, Dong Xu, Benyu Zhang, Hong-jiang Zhang, Qiang Yang, Stephen Lin - IEEE TRANS. PATTERN ANAL. MACH. INTELL , 2007
"... Over the past few decades, a large family of algorithms—supervised or unsupervised; stemming from statistics or geometry theory—has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present in this paper ..."
Abstract - Cited by 271 (29 self) - Add to MetaCart
Over the past few decades, a large family of algorithms—supervised or unsupervised; stemming from statistics or geometry theory—has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present

Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps

by R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, F. Warner, S. Zucker - Proceedings of the National Academy of Sciences , 2005
"... of contexts of data analysis, such as spectral graph theory, manifold learning, nonlinear principal components and kernel methods. We augment these approaches by showing that the diffusion distance is a key intrinsic geometric quantity linking spectral theory of the Markov process, Laplace operators ..."
Abstract - Cited by 257 (45 self) - Add to MetaCart
operators, or kernels, to the corresponding geometry and density of the data. This opens the door to the application of methods from numerical analysis and signal processing to the analysis of functions and transformations of the data. Abstract. We provide a framework for structural multiscale geometric

On graph kernels: Hardness results and efficient alternatives

by Thomas Gärtner, Peter Flach, Stefan Wrobel - IN: CONFERENCE ON LEARNING THEORY , 2003
"... As most ‘real-world’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances tha ..."
Abstract - Cited by 184 (6 self) - Add to MetaCart
that are represented by graphs. So far, only very specific graphs such as trees and strings have been considered. This paper investigates kernels on labeled directed graphs with general structure. It is shown that computing a strictly positive definite graph kernel is at least as hard as solving the graph isomorphism

Combinatorial Auctions with Decreasing Marginal Utilities

by Benny Lehmann, Daniel Lehmann, Noam Nisan , 2001
"... This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross s ..."
Abstract - Cited by 202 (25 self) - Add to MetaCart
substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valuations is NP-hard, we present an efficient greedy 2-approximation algorithm for this case and generalize
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