Discrete Reproducing Kernel Hilbert Spaces: Sampling and Distribution of Dirac-masses
Authors: Palle Jorgensen, Feng Tian
JMLR 2015 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) H over infinite, discrete and countable sets V. In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of V. Illustrations include certain models from neural networks: An Extreme Learning Machine (ELM) is a neural networkconfiguration in which a hidden layer of weights are randomly sampled, and where the object is then to compute resulting output. For RKHSs H of functions defined on a prescribed countable infinite discrete set V , we characterize those which contain the Dirac masses δx for all points x in V. Further examples and applications where this question plays an important role are: (i) discrete Brownian motion-Hilbert spaces, i.e., discrete versions of the Cameron-Martin Hilbert space; (ii) energy-Hilbert spaces corresponding to graph-Laplacians where the set V of vertices is then equipped with a resistance metric; and finally (iii) the study of Gaussian free fields. |
| Researcher Affiliation | Academia | Palle Jorgensen EMAIL Department of Mathematics The University of Iowa Iowa City, IA 52242-1419, U.S.A. Feng Tian EMAIL Department of Mathematics, Informatics, and Cybersecurity Trine University Angola, IN 46703, U.S.A. |
| Pseudocode | No | The paper does not contain any structured pseudocode or algorithm blocks. It is a theoretical mathematics paper focusing on definitions, theorems, lemmas, and proofs. |
| Open Source Code | No | The paper does not contain any concrete access information for source code. It is a theoretical mathematics paper without an implementation-focused methodology. |
| Open Datasets | No | The paper does not provide any concrete access information for publicly available datasets. It is a theoretical paper that does not conduct empirical studies requiring specific datasets. |
| Dataset Splits | No | The paper does not provide specific dataset split information as it is a theoretical work and does not involve experimental evaluation on datasets. |
| Hardware Specification | No | The paper does not provide specific hardware details as it is a theoretical work and does not describe any experimental setup requiring hardware. |
| Software Dependencies | No | The paper does not provide specific ancillary software details as it is a theoretical work and does not describe any computational implementation. |
| Experiment Setup | No | The paper does not contain specific experimental setup details or hyperparameters as it is a theoretical work and does not describe any experiments. |