Searcharxiv⌕ Search

arXiv subjects

Jean Carlo Guella

Publications and source records attributed to Jean Carlo Guella.

10 recordsLinked to original sources
arXiv2026

On symmetric partition lattices and probability interactions

The Möbius function of a partition lattice can be used to define interactions of probability measures. We call a lower set a symmetric partition lattice when it is invariant under permutations of the variables. We characterize these lattices through integer partitions and obtain a recursion for their Möbius coefficients at the top element. By constructing probability measures whose partition products are linearly independent, we show that permutation invariance of an interaction is equivalent to that of its underlying lower set, provided the coordinate spaces are sufficiently large. We also characterize the partitions on a lower set that force an interaction to vanish for every probability measure, and construct indecomposable distributions with zero interaction. We define the order of a symmetric partition lattice and relate it to vanishing marginals. We study generalized Lancaster and define the Streitberg and size-limited partition lattices, obtaining explicit Möbius coefficients for them. Finally, we express interaction equations in terms of characteristic functions. For Gaussian distributions, we show that an interaction vanishes exactly when the distribution factorizes according to a nontrivial partition in the underlying lower set. We also give a nonvanishing criterion for radial characteristic functions generated by Bernstein functions.

math.PR↗
arXiv2026

Hilbert space embeddings of independence tests and interaction measures of several variables

We present a unified theoretical framework for kernel-based measures of dependence on product spaces. Building on the ideas underlying distance covariance, distance multivariance, and the Hilbert-Schmidt Independence Criterion (HSIC), we define a new family of kernels on an $n$-fold Cartesian product, termed positive definite independent of order $k$ (PDI$_{k}$ kernels). These kernels extend the concepts of positive definite and conditionally negative definite kernels to higher orders and provide the foundation for generalized independence and interaction tests, such as the generalized Lancaster interaction of order $k$ ($Λ_{k}^{n}$), and the Streitberg interaction ($Σ$). Our analysis focuses on the continuous setting, where we prove a Kernel Mean Embedding Theorem for PDI$_{k}$ kernels and establish the corresponding integrability restrictions. Based on these results, we characterize how the Kronecker products of PDI kernels behave.

math.FA↗
arXiv2024

Hilbert space embeddings of independence tests of several variables with radial basis functions

In this paper, we characterize several classes of continuous radial basis functions that can be employed to determine whether a interaction of a probability is zero or not. These functions encompass standard independence tests but also the Lancaster/Streitberg interactions, and are multivariate extensions of Bernstein functions. Addressing a gap in these two probability contexts of interactions, we introduce an indexed measure of independence that generalizes the Lancaster interaction. We present several examples of these functions derived from high-order completely monotone functions.

math.CA↗
arXiv2023

Strictly positive definite kernels on compact Riemannian manifolds

The paper studies strictly positive definite kernels on compact Riemannian manifolds. We state new conditions to ensure strict positive definiteness for general kernels and kernels with certain convolutional structure. We also state conditions for such kernels on product manifolds. As an example conditions for products of two-point homogeneous spaces are presented.

math.NA↗
arXiv2022

Positive definite matrix-valued kernels and their scalar valued projections: counterexamples

In this paper we show that the strictly positive definite matrix valued isotropic kernels in the circle and the real dot product kernels in Euclidean spaces are not well behaved with respect to its scalar valued projections. We generalize the counterexamples that we obtained to an abstract setting by using the concepts of unitarily invariant kernels and adjointly invariant kernels, provided the existence of an aperiodic invariant function.

math.CA↗
arXiv2022

Generalization of the HSIC and distance covariance using positive definite independent kernels

Hilbert-Schmidt independence criterion and distance covariance are methods to describe independence of random variables using either the Kronecker product of positive definite kernels or the Kronecker product of conditionally negative definite kernels. In this paper we generalize both methods by providing an independence criteria using a new concept, of positive definite independent kernels. We provide a characterization of the radial kernels that are positive definite independent on all Euclidean spaces and we present several examples.

math.FA↗
arXiv2021

On Gaussian kernels on Hilbert spaces and kernels on Hyperbolic spaces

This paper describes the concepts of Universal/ Integrally Strictly Positive Definite/ $C_{0}$-Universal for the Gaussian kernel on a Hilbert space. As a consequence we obtain a similar characterization for an important family of kernels studied and developed by Schoenberg and also on a family of spatial-time kernels popular on geostatistics, the Gneiting class, and its generalizations. Either by using similar techniques, or by a direct consequence of the Gaussian kernel on Hilbert spaces, we characterize the same concepts for a family of kernels defined on a Hyperbolic space.

math.FA↗
arXiv2021

Generalization of the energy distance by Bernstein functions

We reprove the well known fact that the energy distance defines a metric on the space of Borel probability measures on a Hilbert space with finite first moment by a new approach, by analyzing the behavior of the Gaussian kernel on Hilbert spaces and a Maximum Mean Discrepancy analysis. From this new point of view we are able to generalize the energy distance metric to a family of kernels related to Bernstein functions and conditionally negative definite kernels. We also explain what occurs on the energy distance on the kernel $\|x-y\|^α$ for every $α>2$, where we also generalize the idea to a family of kernels related to derivatives of completely monotone functions and conditionally negative definite kernels.

math.FA↗
arXiv2020

Operator valued positive definite kernels and differentiable universality

We present a characterization for a positive definite operator valued kernel to be universal or $C_{0}$-universal, and apply these characterizations to a family of operator valued kernels that are shown to be well behaved. Later, we obtain a characterization for an operator valued differentiable kernel to be $C^{q}$-universal and $C_{0}^{q}$-universal. In order to obtain such characterization and examples we generalize some well known results concerning the structure of differentiable kernels to the operator valued context. On the examples is given an emphasis on the radial kernels on Euclidean spaces.

math.FA↗