English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Positive-definite kernel functions are fundamental elements of kernel methodsand Gaussian processes. A well-known construction of such functions comes fromBochner's characterization, which connects a positive-definite function with aprobability distribution. Another construction, which appears to have attractedless attention, is Polya's criterion that characterizes a subset of thesefunctions. In this paper, we study the latter characterization and derive anumber of novel kernels little known previously.  In the context of large-scale kernel machines, Rahimi and Recht (2007)proposed a random feature map (random Fourier) that approximates a kernelfunction, through independent sampling of the probability distribution inBochner's characterization. The authors also suggested another feature map(random binning), which, although not explicitly stated, comes from Polya'scharacterization. We show that with the same number of random samples, therandom binning map results in an Euclidean inner product closer to the kernelthan does the random Fourier map. The superiority of the random binning map isconfirmed empirically through regressions and classifications in thereproducing kernel Hilbert space.

On Bochner's and Polya's Characterizations of Positive-Definite Kernels and the Respective Random Feature Maps