2016/01/27 by Palle E. T. Jørgensen, Palle Jorgensen, Feng Tian +2 · 1 citation
Engineering · Mathematics · Medicine · #22E70 #31A15 #31C20 #39A12 #42C15 #46N30 #46N50 #58J65 #62D05 #65R10 #94A20 #Advanced MRI Techniques and Applications #Artificial intelligence #Combinatorics #Computer science #Discrete mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Graph #Hilbert space #Inference #Kernel (algebra) #Mathematics #Medical Imaging Techniques and Applications #Norm (philosophy) #Positive-definite matrix #Primary 47L60 #Probability (math.PR) #Pure mathematics #Reproducing kernel Hilbert space #Secondary 46N20 #Sparse and Compressive Sensing Techniques #Spectral Theory (math.SP) #math.FA #math.PR #math.SP #msc:22E70 #msc:31A15 #msc:31C20 #msc:39A12 #msc:42C15 #msc:46N20 #msc:46N30 #msc:46N50 #msc:47L60 #msc:58J65 #msc:62D05 #msc:65R10 #msc:94A20
paper · pdf · doi:10.48550/arxiv.1601.07380
published in arXiv (Cornell University) (Cornell University) · arXiv admin note: substantial text overlap with arXiv:1501.02310
arxiv created 2016/01/27 · openalex publication_date 2016/01/27 · arxiv updated 2016/01/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In a general context of positive definite kernels k, we develop tools and algorithms for sampling in reproducing kernel Hilbert space \mathscrH (RKHS). With reference to these RKHSs, our results allow inference from samples; more precisely, reconstruction of an "entire" (or global) signal, a function f from \mathscrH, via generalized interpolation of f from partial information obtained from carefully chosen distributions of sample points. We give necessary and sufficient conditions for configurations of point-masses δx of sample-points x to have finite norm relative to the particular RKHS \mathscrH considered. When this is the case, and the kernel k is given, we obtain an induced positive definite kernel ⟨ δx,δy⟩ _\mathscrH. We perform a comparison, and we study when this induced positive definite kernel has l2 rows and columns. The latter task is accomplished with the use of certain symmetric pairs of operators in the two Hilbert spaces, l2 on one side, and the RKHS \mathscrH on the other. A number of applications are given, including to infinite network systems, to graph Laplacians, to resistance metrics, and to sampling of Gaussian fields.