[Paper Review] Strict positive definiteness on a product of compact two-point homogeneous spaces

V. S. Barbosa, V‎. ‎A‎. Menegatto|arXiv (Cornell University)|May 23, 2016

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper provides a complete characterization of strictly positive definite, continuous, isotropic kernels on products of compact two-point homogeneous spaces, focusing on cases where at least one factor is a sphere of dimension greater than one and the other is not a circle. Using series expansions involving Jacobi polynomials and convergence conditions, the authors establish necessary and sufficient conditions for strict positive definiteness based on the distribution of expansion coefficients across even and odd degrees in the spectral decomposition.

ABSTRACT

We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, in the cases in which at least one of the spaces is a sphere of dimension greater than 1 and the other is not a circle. The result complements similar characterizations previously obtained for products of high dimensional spheres.

Motivation & Objective

• To characterize real, continuous, isotropic, and strictly positive definite kernels on products of compact two-point homogeneous spaces.
• To extend existing results on strict positive definiteness from products of high-dimensional spheres to more general spaces.
• To establish necessary and sufficient conditions for strict positive definiteness in terms of spectral coefficients when one factor is a sphere (dim ≥ 2) and the other is not a circle.
• To address the gap in the literature regarding strict positive definiteness on mixed products involving spheres and other two-point homogeneous spaces (e.g., projective spaces).
• To provide a unified framework using hypergeometric functions and orthogonal polynomial expansions for analyzing positive definiteness on such product spaces.

Proposed method

• Utilizes the isotropy condition to express kernels on product spaces as functions of geodesic distances via the cosine of half-distances.
• Employs a series expansion of the isotropic part in terms of Jacobi polynomials $ P_k^{(d-2)/2,eta}(t) $ and $ P_l^{(d'-2)/2,eta'}(s) $, with non-negative coefficients.
• Applies convergence criteria involving the evaluation of Jacobi polynomials at 1 to ensure positive definiteness.
• Introduces the concept of $ DC $-strict positive definiteness and uses it as a stepping stone to derive conditions for full strict positive definiteness.
• Applies spectral analysis and functional equations involving orthogonal polynomial systems to analyze the vanishing of coefficient sequences.
• Uses contradiction and limit arguments on coefficient sequences to prove that the absence of infinite sequences of even and odd degrees in the support of the coefficient set implies failure of strict positive definiteness.

Experimental results

Research questions

• RQ1 What conditions on the spectral coefficients of an isotropic kernel ensure strict positive definiteness on a product of compact two-point homogeneous spaces when one factor is a sphere of dimension ≥ 2 and the other is not a circle?
• RQ2 How does the distribution of coefficients across even and odd degrees in the Jacobi polynomial expansion relate to strict positive definiteness on such product spaces?
• RQ3 Can the characterization of strict positive definiteness on $ S^d imes ext{other} $ spaces be extended beyond the case of two spheres?
• RQ4 What role do antipodal sets and geodesic distance properties play in the analysis of strict positive definiteness on these manifolds?
• RQ5 Is it possible to generalize the known results on strict positive definiteness for products of spheres to products involving projective spaces or other two-point homogeneous spaces?

Key findings

• A real, continuous, isotropic kernel $ K $ on $ ext{M}^d imes ext{H}^{d'} $ is positive definite if and only if its isotropic part admits a series expansion in Jacobi polynomials with non-negative coefficients and finite total mass.
• For $ K $ to be strictly positive definite on $ S^d imes ext{H}^{d'} $ with $ d e 1 $ and $ ext{H}^{d'} $ not a sphere, it is necessary and sufficient that the coefficient set $ J_K $ contains two sequences: one with infinitely many even $ k $-indices and one with infinitely many odd $ k $-indices, both diverging to infinity.
• The strict positive definiteness of $ K $ is equivalent to the non-vanishing of the coefficient sequences in both even and odd degree components across the spectral decomposition.
• The proof relies on showing that the only solution to the system of equations derived from the kernel evaluation is trivial when such infinite sequences exist, ensuring positive definiteness of all interpolation matrices.
• The characterization extends previous results on $ S^d imes S^{d'} $ to broader classes of compact two-point homogeneous spaces, excluding cases involving $ S^1 $.
• The open problem remains for products involving $ S^1 $, particularly when one or both factors are circles.