Emilio Porcu

Simon Hubbert, Emilio Porcu, Chris. J. Oates et al.

This work provides theoretical foundations for kernel methods in the hyperspherical context. Specifically, we characterise the native spaces (reproducing kernel Hilbert spaces) and the Sobolev spaces associated with kernels defined over hyperspheres. Our results have direct consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method. We first introduce a suitable characterisation on Sobolev spaces on the $d$-dimensional hypersphere embedded in $(d+1)$-dimensional Euclidean spaces. Our characterisation is based on the Fourier--Schoenberg sequences associated with a given kernel. Such sequences are hard (if not impossible) to compute analytically on $d$-dimensional spheres, but often feasible over Hilbert spheres. We circumvent this problem by finding a projection operator that allows to Fourier mapping from Hilbert into finite dimensional hyperspheres. We illustrate our findings through some parametric families of kernels.

Xavier Emery, Emilio Porcu, Moreno Bevilacqua

There exists a plethora of parametric models for positive definite kernels, and their use is ubiquitous in disciplines as diverse as statistics, machine learning, numerical analysis, and approximation theory. Usually, the kernel parameters index certain features of an associated process. Amongst those features, smoothness (in the sense of Sobolev spaces, mean square differentiability, and fractal dimensions), compact or global supports, and negative dependencies (hole effects) are of interest to several theoretical and applied disciplines. This paper unifies a wealth of well-known kernels into a single parametric class that encompasses them as special cases, attained either by exact parameterization or through parametric asymptotics. We furthermore characterize the Sobolev space that is norm equivalent to the RKHS associated with the new kernel. As a by-product, we infer the Sobolev spaces that are associated with existing classes of kernels. We illustrate the main properties of the new class, show how this class can switch from compact to global supports, and provide special cases for which the kernel attains negative values over nontrivial intervals. Hence, the proposed class of kernel is the reproducing kernel of a very rich Hilbert space that contains many special cases, including the celebrated Matérn and Wendland kernels, as well as their aliases with hole effects.

Emilio Porcu, Roy El Moukari, Laurent Najman et al.

This manuscript provides a holistic (data-centric) view of what we term essential data science, as a natural ecosystem with challenges and missions stemming from the data universe with its multiple combinations of the 5D complexities (data structure, domain, cardinality, causality, and ethics) with the phases of the data life cycle. Data agents perform tasks driven by specific goals. The data scientist is an abstract entity that comes from the logical organization of data agents with their actions. Data scientists face challenges that are defined according to the missions. We define specific discipline-induced data science, which in turn allows for the definition of pan-data science, a natural ecosystem that integrates specific disciplines with the essential data science. We semantically split the essential data science into computational, and foundational. We claim that there is a serious threat of divergence between computational and foundational data science. Especially, if no approach is taken to rate whether a data universe discovery should be useful or not. We suggest that rigorous approaches to measure the usefulness of data universe discoveries might mitigate such a divergence.