Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInterpolating splines on graphs for data science applications
98
0
0.0
(
0
)
Added by
John Paul Ward
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic Lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the Lagrange function, showing that they scale with the density of the data. These results indicate that Lagrange-type bases are ideal building blocks for analyzing data on graphs, and we illustrate their use in kernel-based machine learning applications.
rate research
Read More
Lagrange functions are localized bases that have many applications in signal processing and data approximation. Their structure and fast decay make them excellent tools for constructing approximations. Here, we propose perturbations of Lagrange functions on graphs that maintain the nice properties of Lagrange functions while also having the added benefit of being locally supported. Moreover, their local construction means that they can be computed in parallel, and they are easily implemented via quasi-interpolation.
A generalized spline on a graph $G$ with edges labeled by ideals in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the edge-labelings consist of at most two finitely-generated ideals, and $2)$ for cycles when the edge-labelings consist of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in classical (analytic) splines.
In this paper, we study how to quickly compute the <-minimal monomial interpolating basis for a multivariate polynomial interpolation problem. We address the notion of reverse reduced basis of linearly independent polynomials and design an algorithm for it. Based on the notion, for any monomial ordering we present a new method to read off the <-minimal monomial interpolating basis from monomials appearing in the polynomials representing the interpolation conditions.
Differentiable real function reproducing primes up to a given number and having a differentiable inverse function is constructed. This inverse function is compared with the Riemann-Von Mangoldt exact expression for the number of primes not exceeding a given value. Software for computation of the direct and inverse functions and their derivatives is developed. Examples of approximate solution of Diophantine equations on the primes are given.
In this paper, we demonstrate the construction of generalized Rough Polyhamronic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on edge or derivative measurements. We prove the (quasi)-optimal localization and approximation properties of the obtained bases, and justify the theoretical results with numerical experiments.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
