Results 1  10
of
22
Convolution Kernels on Discrete Structures
, 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 ..."
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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 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, pairHMMs, or ANOVA decompositions. Uses of the method lead to open problems involving the theory of infinitely divisible positive definite functions. Fundamentals of this theory and the theory of reproducing kernel Hilbert spaces are reviewed and applied in establishing the validity of the method.
CauchySchwarz Inequalities Associated with Positive Semidefinite Matrices
, 2000
"... Using a quasilinear representation for unitarily invariant norms, we prove a basic inequality: Let A = ` L X X ..."
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Cited by 15 (7 self)
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Using a quasilinear representation for unitarily invariant norms, we prove a basic inequality: Let A = ` L X X
Monotonicity for entrywise functions of matrices
, 709
"... We characterize real functions f on an interval (−α, α) for which the entrywise matrix function [aij] ↦ → [f(aij)] is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown. ..."
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Cited by 8 (0 self)
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We characterize real functions f on an interval (−α, α) for which the entrywise matrix function [aij] ↦ → [f(aij)] is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown.
Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity
, 2013
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FourierStieltjes algebras of rdiscrete groupoids
 J. OPERATOR THEORY
, 1999
"... The FourierStieltjes algebra, B(G), of a groupoid G has recently been defined using suitably defined positive definite functions. In this paper we prove various properties of positive definite functions including that continuous positive definite functions separate the points of G. We also show th ..."
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Cited by 4 (0 self)
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The FourierStieltjes algebra, B(G), of a groupoid G has recently been defined using suitably defined positive definite functions. In this paper we prove various properties of positive definite functions including that continuous positive definite functions separate the points of G. We also show that in certain cases the continuous elements of B(G) (denoted B(G)) and the space of complexvalued bounded continuous functions (denoted C(G)) are topologically isomorphic as Banach algebras but not as ordered or ∗Banach algebras. The same is shown to be true for B(G) and the space of complexvalued bounded Borel functions (denoted M(G)). We explore various conditions including that of an ordering map that one can place on groupoids and their connections, and we show that if G has such an ordering
Preserving positivity for rankconstrained matrices
 Trans. Amer. Math. Soc
, 2015
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THE CRITICAL EXPONENT FOR CONTINUOUS CONVENTIONAL POWERS OF DOUBLY NONNEGATIVE MATRICES
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ON FRACTIONAL HADAMARD POWERS OF POSITIVE BLOCK MATRICES
"... Abstract. Entrywise powers of matrices have been wellstudied in the literature, and have recently received renewed attention due to their application in the regularization of highdimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices (Hst) n s,t=1 ..."
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Abstract. Entrywise powers of matrices have been wellstudied in the literature, and have recently received renewed attention due to their application in the regularization of highdimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices (Hst) n s,t=1 where each block Hst is a complex m × m matrix. We first characterize the powers α ∈ R such that the blockwise power map (Hst) 7 → (H α st) preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions which was developed by C. Loewner. Second, we revisit previous work by D. Choudhury [Proc. Amer. Math. Soc. 108] who had provided a lower bound on α for preserving positivity when the blocks Hst pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous results by FitzGeraldHorn, BhatiaElsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the cases where the blocks Hst are diagonalizable consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given β ∈ Z, we provide lower and upper bounds for the threshold power α> 0 above which the complex characters z = reiθ 7 → rαeiβθ preserve positivity when applied entrywise to Hermitian positive semidefinite matrices. In particular, we completely resolve the n = 3 case of a question raised in 2001 by Xingzhi Zhan. As an application of our results, we also extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters K of the complex plane for which the map (Hst) n s,t=1 7 → (K(tr(Hst)))
A Stochastic Uncoupling Process for Graphs
 NATIONAL RESEARCH INSTITUTE FOR MATHEMATICS AND COMPUTER SCIENCE IN THE
, 2000
"... A discrete stochastic uncoupling process for finite spaces is introduced, called the Markov Cluster Process. The process takes a stochastic matrix as input, and then alternates flow expansion and flow inflation, each step defining a stochastic matrix in terms of the previous one. Flow expansion corr ..."
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Cited by 2 (1 self)
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A discrete stochastic uncoupling process for finite spaces is introduced, called the Markov Cluster Process. The process takes a stochastic matrix as input, and then alternates flow expansion and flow inflation, each step defining a stochastic matrix in terms of the previous one. Flow expansion corresponds with taking the k th power of a stochastic matrix, where k 2 IN . Flow inflation corresponds with a parametrized operator \Gamma r , r 0, which maps the set of (column) stochastic matrices onto itself. The image \Gamma r M is obtained by raising each entry in M to the r th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit which is idempotent under both matrix multiplication and inflation, with quadratic convergence around the limit points. The limit is in general extremely sparse and the number of components of its associated graph may be larger than the number associated with the input matrix. This uncoupli...