Online regularized pairwise learning with least squares loss

2019 ◽  
Vol 18 (01) ◽  
pp. 49-78 ◽  
Author(s):  
Cheng Wang ◽  
Ting Hu

In this paper, we study online algorithm for pairwise problems generated from the Tikhonov regularization scheme associated with the least squares loss function and a reproducing kernel Hilbert space (RKHS). This work establishes the convergence for the last iterate of the online pairwise algorithm with the polynomially decaying step sizes and varying regularization parameters. We show that the obtained error rate in [Formula: see text]-norm can be nearly optimal in the minimax sense under some mild conditions. Our analysis is achieved by a sharp estimate for the norms of the learning sequence and the characterization of RKHS using its associated integral operators and probability inequalities for random variables with values in a Hilbert space.

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jia Cai

We investigate a coefficient-based least squares regression problem with indefinite kernels from non-identical unbounded sampling processes. Here non-identical unbounded sampling means the samples are drawn independently but not identically from unbounded sampling processes. The kernel is not necessarily symmetric or positive semi-definite. This leads to additional difficulty in the error analysis. By introducing a suitable reproducing kernel Hilbert space (RKHS) and a suitable intermediate integral operator, elaborate analysis is presented by means of a novel technique for the sample error. This leads to satisfactory results.


2018 ◽  
Vol 311 ◽  
pp. 235-244 ◽  
Author(s):  
Xiang-Jun Shen ◽  
Yong Dong ◽  
Jian-Ping Gou ◽  
Yong-Zhao Zhan ◽  
Jianping Fan

Author(s):  
Michael T Jury ◽  
Robert T W Martin

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.


Author(s):  
Dominic Knoch ◽  
Christian R. Werner ◽  
Rhonda C. Meyer ◽  
David Riewe ◽  
Amine Abbadi ◽  
...  

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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