Harmonic Analysis on Semigroups

Berg, Christian; Christensen, Jens Peter; Ressel, Paul

doi:10.1007/978-1-4612-1128-0

Cited by 895 publications

(572 citation statements)

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“…However, the relationship between the couple (w, ρ) and the bag of path being not bijective, the final kernel between bags is only semi positive-definite [13]. Though, in all our experiments run so far the Gram matrices associated to the bags of paths were positive-definite.…”

Section: Change Detection Kernelmentioning

confidence: 91%

Edition within a Graph Kernel Framework for Shape Recognition

Dupé¹,

Brun²

2009

Graph-Based Representations in Pattern Recognition

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To cite this version:François-Xavier Dupé, Luc Brun. Edition within a graph kernel framework for shape recognition. Graph Based Representation in Pattern Recognition 2009, 2009, Venice, Italy. pp.11-20, 2009 Abstract. A large family of shape comparison methods is based on a medial axis transform combined with an encoding of the skeleton by a graph. Despite many qualities this encoding of shapes suffers from the non continuity of the medial axis transform. In this paper, we propose to integrate robustness against structural noise inside a graph kernel. This robustness is based on a selection of the paths according to their relevance and on path editions. This kernel is positive semi-definite and several experiments prove the efficiency of our approach compared to alternative kernels.

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Section: Change Detection Kernelmentioning

confidence: 91%

Edition within a Graph Kernel Framework for Shape Recognition

Dupé¹,

Brun²

2009

Graph-Based Representations in Pattern Recognition

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“…This class was introduced and extensively studied by S. Bernstein [5] and D. Widder [30] in connection with the so-called completely (or absolutely) monotonic analytic functions (see the definition in Section 3.2). We only mention a deep penetration of the both classes into complex analysis, inequalities analysis [2], special functions [25], probability theory [19], radial-function interpolation [29], harmonic analysis on semigroups [3] (for further discussion and references, see recent survey [4]). …”

Section: ])mentioning

confidence: 99%

Length functions of lemniscates

Kuznetsova

Tkachev²

2003

manuscripta mathematica

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Abstract. We study metric and analytic properties of generalized lemniscates Et(f ) = {z ∈ C : ln |f (z)| = t}, where f is an analytic function. Our main result states that the length function |Et(f )| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln |Et(f )| is convex on any interval free of critical points of ln |f |. As another application we deduce explicit formulae of the length function in some special cases.

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“…Critchley suggests the power transformation f (δ i j ) = δ µ i j (for µ > 0). If the dissimilarities are derived from Euclidean distances, we note that the kernel k(x, y) = − x − y β is conditionally positive definite (CPD) if β ≤ 2 (Berg, Christensen, & Ressel, 1984). When the kernel is CPD, the centered matrix will be positive definite.…”

Section: Metric Mdsmentioning

confidence: 99%

Untitled

Williams

2002

Machine Learning

128

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Abstract. In this note we show that the kernel PCA algorithm of Schölkopf, Smola, and Müller (Neural Computation, 10, 1299-1319.) can be interpreted as a form of metric multidimensional scaling (MDS) when the kernel function k(x, y) is isotropic, i.e. it depends only on x − y . This leads to a metric MDS algorithm where the desired configuration of points is found via the solution of an eigenproblem rather than through the iterative optimization of the stress objective function. The question of kernel choice is also discussed.

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Harmonic Analysis on Semigroups

Cited by 895 publications

References 0 publications

Edition within a Graph Kernel Framework for Shape Recognition

Edition within a Graph Kernel Framework for Shape Recognition

Length functions of lemniscates

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