A LEARNING-BASED FRAMEWORK FOR GRAPH MATCHING

2004 ◽  
Vol 18 (03) ◽  
pp. 355-374 ◽  
Author(s):  
M. A. VAN WYK ◽  
B. J. VAN WYK
Keyword(s):  
Reproducing Kernel ◽  
Graph Matching ◽  
Reproducing Kernel Hilbert Space ◽  
Explicit Calculation ◽  
Input Output ◽  
Attributed Graph ◽  
Special Cases ◽  
Variable Mapping ◽  
Improved Performance ◽  
Feature Augmentation

This paper presents a unifying review of a learning-based framework for kernel-based attributed graph matching. The framework, which includes as special cases the RKHS Interplator-Based Graph Matching (RIGM) and Interpolator-Based Kronecker Product Graph Matching (IBKPGM) algorithms, incorporates a general approach where no assumption is made about the adjacency structure of the graphs to be matched. Corresponding pairs of attributed adjacency matrices and attribute vectors of an input and reference graph are used as the input–output training set of a constrained multi-input multi-output multi-variable mapping to be learned. It is shown that a Reproducing Kernel Hilbert Space (RKHS) based interpolator can be used to infer this mapping. Partially constraining the inferred mapping by the generation of additional consistency input–output training pairs and the use of polynomial feature augmentation lead to improved performance. The proposed learning-based framework avoids the explicit calculation of compatibility values.

How to Incorporate New Pattern Pairs without Having to Teach the Previously Acquired Pattern Pairs

1992 ◽  
Vol 4 (6) ◽  
pp. 880-887 ◽  
Author(s):  
H. M. Wabgaonkar ◽  
A. R. Stubberud
Keyword(s):  
Hilbert Space ◽  
Associative Memory ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Input Output ◽  
Space Setting ◽  
Output Pattern ◽  
Orthogonalization Procedure ◽  
Memory Synthesis

In this paper, we deal with the problem of associative memory synthesis. The particular issue that we wish to explore is the ability to store new input-output pattern pairs without having to modify the path weights corresponding to the already taught pattern pairs. The approach to the solution of this problem is via interpolation carried out in a Reproducing Kernel Hilbert Space setting. An orthogonalization procedure carried out on a properly chosen set of functions leads to the solution of the problem.

scholarly journals Detecting periodicities with Gaussian processes

2016 ◽  
Vol 2 ◽  
pp. e50 ◽  
Author(s):  
Nicolas Durrande ◽  
James Hensman ◽  
Magnus Rattray ◽  
Neil D. Lawrence
Keyword(s):  
Hilbert Space ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Gaussian Process Regression ◽  
Inner Product ◽  
Input Output ◽  
Periodic Data

We consider the problem of detecting and quantifying the periodic component of a function given noise-corrupted observations of a limited number of input/output tuples. Our approach is based on Gaussian process regression, which provides a flexible non-parametric framework for modelling periodic data. We introduce a novel decomposition of the covariance function as the sum of periodic and aperiodic kernels. This decomposition allows for the creation of sub-models which capture the periodic nature of the signal and its complement. To quantify the periodicity of the signal, we derive a periodicity ratio which reflects the uncertainty in the fitted sub-models. Although the method can be applied to many kernels, we give a special emphasis to the Matérn family, from the expression of the reproducing kernel Hilbert space inner product to the implementation of the associated periodic kernels in a Gaussian process toolkit. The proposed method is illustrated by considering the detection of periodically expressed genes in thearabidopsisgenome.

scholarly journals Detecting periodicities with Gaussian processes

2016 ◽  
Author(s):  
Nicolas Durrande ◽  
James Hensman ◽  
Magnus Rattray ◽  
Neil D Lawrence
Keyword(s):  
Hilbert Space ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Gaussian Process Regression ◽  
Inner Product ◽  
Input Output ◽  
Periodic Data

We consider the problem of detecting and quantifying the periodic component of a function given noise-corrupted observations of a limited number of input/output tuples. Our approach is based on Gaussian process regression which provides a flexible non-parametric framework for modelling periodic data. We introduce a novel decomposition of the covariance function as the sum of periodic and aperiodic kernels. This decomposition allows for the creation of sub-models which capture the periodic nature of the signal and its complement. To quantify the periodicity of the signal, we derive a periodicity ratio which reflects the uncertainty in the fitted sub-models. Although the method can be applied to many kernels, we give a special emphasis to the Matérn family, from the expression of the reproducing kernel Hilbert space inner product to the implementation of the associated periodic kernels in a Gaussian process toolkit. The proposed method is illustrated by considering the detection of periodically expressed genes in the arabidopsis genome.

scholarly journals Detecting periodicities with Gaussian processes

2016 ◽  
Author(s):  
Nicolas Durrande ◽  
James Hensman ◽  
Magnus Rattray ◽  
Neil D Lawrence
Keyword(s):  
Hilbert Space ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Gaussian Process Regression ◽  
Inner Product ◽  
Input Output ◽  
Periodic Data

We consider the problem of detecting and quantifying the periodic component of a function given noise-corrupted observations of a limited number of input/output tuples. Our approach is based on Gaussian process regression which provides a flexible non-parametric framework for modelling periodic data. We introduce a novel decomposition of the covariance function as the sum of periodic and aperiodic kernels. This decomposition allows for the creation of sub-models which capture the periodic nature of the signal and its complement. To quantify the periodicity of the signal, we derive a periodicity ratio which reflects the uncertainty in the fitted sub-models. Although the method can be applied to many kernels, we give a special emphasis to the Matérn family, from the expression of the reproducing kernel Hilbert space inner product to the implementation of the associated periodic kernels in a Gaussian process toolkit. The proposed method is illustrated by considering the detection of periodically expressed genes in the arabidopsis genome.

scholarly journals Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi ◽  
D. Baleanu
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Functional Differential Equations ◽  
Variational Iteration Method ◽  
Proportional Delay ◽  
Neutral Functional Differential Equations ◽  
Special Cases ◽  
Functional Differential ◽  
Pseudo Spectral

The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-legθ-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

scholarly journals Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

scholarly journals A New Upper Bound for Sampling Numbers

2021 ◽  
Author(s):  
Nicolas Nagel ◽  
Martin Schäfer ◽  
Tino Ullrich
Keyword(s):  
Upper Bound ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Procedure ◽  
Compact Embedding ◽  
Recovery Method ◽  
Grid Sampling ◽  
Integrable Functions ◽  
Random Draw ◽  
Square Integrable Functions

AbstractWe provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

scholarly journals Multi-omics-based prediction of hybrid performance in canola

2021 ◽  
Author(s):  
Dominic Knoch ◽  
Christian R. Werner ◽  
Rhonda C. Meyer ◽  
David Riewe ◽  
Amine Abbadi ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Genetic Markers ◽  
Reproducing Kernel ◽  
Prediction Models ◽  
Agronomic Traits ◽  
Reproducing Kernel Hilbert Space ◽  
Hybrid Performance ◽  
Hybrid Prediction ◽  
Gaussian Kernels ◽  
Breeding Programmes

Abstract Key message Complementing or replacing genetic markers with transcriptomic data and use of reproducing kernel Hilbert space regression based on Gaussian kernels increases hybrid prediction accuracies for complex agronomic traits in canola. In plant breeding, hybrids gained particular importance due to heterosis, the superior performance of offspring compared to their inbred parents. Since the development of new top performing hybrids requires labour-intensive and costly breeding programmes, including testing of large numbers of experimental hybrids, the prediction of hybrid performance is of utmost interest to plant breeders. In this study, we tested the effectiveness of hybrid prediction models in spring-type oilseed rape (Brassica napus L./canola) employing different omics profiles, individually and in combination. To this end, a population of 950 F1 hybrids was evaluated for seed yield and six other agronomically relevant traits in commercial field trials at several locations throughout Europe. A subset of these hybrids was also evaluated in a climatized glasshouse regarding early biomass production. For each of the 477 parental rapeseed lines, 13,201 single nucleotide polymorphisms (SNPs), 154 primary metabolites, and 19,479 transcripts were determined and used as predictive variables. Both, SNP markers and transcripts, effectively predict hybrid performance using (genomic) best linear unbiased prediction models (gBLUP). Compared to models using pure genetic markers, models incorporating transcriptome data resulted in significantly higher prediction accuracies for five out of seven agronomic traits, indicating that transcripts carry important information beyond genomic data. Notably, reproducing kernel Hilbert space regression based on Gaussian kernels significantly exceeded the predictive abilities of gBLUP models for six of the seven agronomic traits, demonstrating its potential for implementation in future canola breeding programmes.

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