Characteristic Kernels and Infinitely Divisible Distributions
Authors: Yu Nishiyama, Kenji Fukumizu
JMLR 2016 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | First, we show, using the L evy Khintchine formula, that any shift-invariant kernel given by a bounded, continuous, and symmetric probability density function (pdf) of an infinitely divisible distribution on Rd is characteristic. We mention some closure properties of such characteristic kernels under addition, pointwise product, and convolution. Second, in developing various kernel mean algorithms, it is fundamental to compute the following values: (i) kernel mean values m P (x), x X, and (ii) kernel mean RKHS inner products m P , m Q H, for probability measures P, Q. If P, Q, and kernel k are Gaussians, then the computation of (i) and (ii) results in Gaussian pdfs that are tractable. We generalize this Gaussian combination to more general cases in the class of infinitely divisible distributions. We then introduce a conjugate kernel and a convolution trick, so that the above (i) and (ii) have the same pdf form, expecting tractable computation at least in some cases. |
| Researcher Affiliation | Academia | Yu Nishiyama EMAIL The University of Electro-Communications 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan Kenji Fukumizu EMAIL The Institute of Statistical Mathematics 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan |
| Pseudocode | No | The paper describes mathematical concepts, theorems, and properties related to characteristic kernels and infinitely divisible distributions, but it does not contain any structured pseudocode or algorithm blocks. |
| Open Source Code | No | The paper mentions 'The STABLE 5.110 software' and 'an R package software called ghyp' as existing tools for computation, but there is no explicit statement from the authors about releasing their own source code for the methodology described in this paper, nor is a specific repository link provided. |
| Open Datasets | No | The paper is theoretical, focusing on mathematical properties and proofs of characteristic kernels and infinitely divisible distributions. It does not describe any experiments that would require a dataset, and therefore, no datasets are mentioned as publicly available or open. |
| Dataset Splits | No | As a theoretical paper, this work does not involve empirical experiments or dataset analysis. Consequently, there is no mention of training, testing, or validation dataset splits. |
| Hardware Specification | No | The paper is theoretical and does not describe any experimental setup or results that would require specific hardware specifications. Therefore, no hardware details are provided. |
| Software Dependencies | Yes | The STABLE 5.110 software allows the computation of α-stable pdfs when they are independent, isotropic, elliptical, or have discrete spectral measures Γd under some settings. More information can be found in the STABLE 5.1 software manual. ... For example, there is an R package software called ghyp on the GH distributions (Breymann and L uthi, 2013). |
| Experiment Setup | No | This paper is theoretical in nature, presenting mathematical frameworks and proofs. It does not include any experimental results, and thus, there are no details regarding experimental setup, hyperparameters, or system-level training settings. |