Kernel Hilbert Spaces

In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. Motivated by our recent algorithm for solving the BAP problem [Censor, Mansour, Reem, J. Approx. Theory (2024)], we discuss the problem in more general normed spaces and with possibly non-convex subsets, and focus our attention on the fundamental issues of uniqueness and existence of the solution to the problem. We present several sufficient geometric conditions for the (at most) uniqueness of a BAP. These conditions are related to the structure and the relative orientation of the boundaries of the subsets and to the norm. We also present many sufficient conditions for the existence of a BAP. In general, the paper re-examines several aspects related to the BAP problem, including the historical one, and shows, probably for the first time, how wide is the scope of the BAP problem in terms of the scientific communities which are involved in it (frequently independently) and in terms of its applications.

Kernel herding belongs to a family of deterministic quadratures that seek to minimize the maximum mean discrepancy (MMD), that is, the worst-case integration error over a reproducing kernel Hilbert space (RKHS). These MMD minimization procedures come with strong experimental support, but comparatively less theoretical footing. In particular, apart from recent progress in distribution compression, little has been proved in favor of an improvement of MMD minimization over classical Monte Carlo quadrature when the RKHS is infinite-dimensional. In this paper, we study a joint probability distribution over quadrature nodes, a tailored Gibbs distribution, whose support intuitively tends to concentrate around MMD minimizers as a temperature parameter is decreased. Our main contribution is to prove that drawing integration nodes from our distribution does outperform i.i.d Monte Carlo. While our bounds on the worst-case integration error feature the same rate as i.i.d. Monte Carlo, we do obtain a tighter concentration inequality as the temperature parameter decreases. This means smaller confidence intervals as the number of quadrature nodes increases. While arguably a first step, our results demonstrate that the mathematical toolbox developed around Gibbs measures can help understand to what extent kernel herding and its variants improve on computationally cheaper methods. There remains the issue of sampling from our Gibbs distribution. In our numerical experiments, we demonstrate that a simple MCMC chain already yields approximate samples that lead to improved confidence intervals around the target integrals, as supported by our theoretical results.

The purpose of this paper is two-fold. First, we show that, after a specified form isometry, the two-coordinate reduction in the binary Hilbert-symbol realization of Chinburg and Zhang is inverse to Kim's building-up construction, up to permutation equivalence. Second, for $q\equiv1\pmod4$, we develop a $q$-ary analogue of this reduction-and-extension mechanism. The identity $c^2=-1$ yields the isotropic line governing the split construction. For every fixed ordered pairing of the coordinates, we obtain a universal rank-$r$ boxed normal form, where $r$ is the dimension of the intersection with the product of these isotropic lines. Applications include optimal self-dual $[6,3,4]$ and $[8,4,4]$ codes over $\mathbb F_{5}$, optimal self-dual $[8,4,5]$ and $[10,5,6]$ codes over $\mathbb F_{13}$, and a self-dual $[12,6,6]$ code over $\mathbb F_{13}$. We also give an exact repeated boxed realization of self-dual $[18,9,8]$ and $[20,10,10]$ codes over $\mathbb F_{13}$, in which the split-boxed parent and its building-up child occur in one complete generator matrix. The algebraic core is formalized in Lean 4.