Related papers: A functional view of upper bounds on codes
We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte's linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal…
Delsarte's method and its extensions allow to consider the upper bound problem for codes in 2-point-homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that…
We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders…
Pfender \textit{[J. Combin. Theory Ser. A, 2007]} provided a one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes, which offers an upper bound for the celebrated…
We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a…
Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…
Two central objects in constructive approximation, the Christoffel-Darboux kernel and the Christoffel function, are encoding ample information about the associated moment data and ultimately about the possible generating measures. We…
A central and longstanding open problem in coding theory is the rate-versus-distance trade-off for binary error-correcting codes. In a seminal work, Delsarte introduced a family of linear programs establishing relaxations on the size of…
Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization…
Bleher and Kuijlaars, and Daems and Kuijlaars showed that the correlation functions of the eigenvalues of a random matrix from unitary ensemble with external source can be expressed in terms of the Christoffel-Darboux kernel for multiple…
We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if…
We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
A method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random (possibly correlated) Jacobi matrix. As an application, one obtains sharp upper bounds…
The Christoffel-Darboux kernels for orthogonal polynomials in several real variables are investigated within the context of the three term recurrence relation reformulated for this purpose. Examples of orthogonal polynomials on the unit…
New upper bounds on the pointwise behaviour of Christoffel function on convex domains in ${\mathbb{R}}^d$ are obtained. These estimates are established by explicitly constructing the corresponding "needle"-like algebraic polynomials having…
Polynomial, or Delsarte's, method in coding theory accounts for a variety of structural results on, and bounds on the size of, extremal configurations (codes and designs) in various metric spaces. In recent works of the authors the…
We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…
Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel-Darboux…
The problem of establishing out-of-sample bounds for the values of an unkonwn ground-truth function is considered. Kernels and their associated Hilbert spaces are the main formalism employed herein along with an observational model where…