Related papers: On multi-dimensional sampling and interpolation
We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the…
We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular,…
Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set…
Many, if not most, inflationary models predict the power-law index of the spectrum of density perturbations is close to one, though not precisely equal to one, |n-1| \sim O(0.1), implying that the spectrum of density perturbations is…
In this paper, I proof that Importance Sampling estimates based on dependent sample sets are consistent under certain conditions. This can be used to reduce variance in Bayesian Models with factorizing likelihoods, using sample sets that…
In the era of big data, we first need to manage the data, which requires us to find missing data or predict the trend, so we need operations including interpolation and data fitting. Interpolation is a process to discover deducing new data…
This work investigates theoretically the interplay between interpolation and aggregation in regression. We establish that the $\gamma$-graph dimension characterizes learnability for a broad class of natural aggregation procedures.…
The Central Limit Theorem provides a foundation for inferential statistics and hypothesis testing. It describes how standardized statistics behave under repeated sampling from large populations. However, if the size of the sample (n)…
Let $S$ be a finite set, and $X_1,\ldots,X_n$ an i.i.d. uniform sample from $S$. To estimate the size $|S|$, without further structure, one can wait for repeats and use the birthday problem. This requires a sample size of the order…
We study the approximation of maps into complex manifolds along with interpolation on certain compact subsets of the plane. Results are also obtained regarding approximation and interpolation of sections of holomorphic submersions.
Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo…
We discuss a generalization of the conditions of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the $L^2$ metric structure of the Gaussian…
In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be…
The Ingleton inequality is a classical linear information inequality that holds for representable matroids but fails to be universally valid for entropic vectors. Understanding the extent to which this inequality can be violated has been a…
The functional interpolation problem on a continual set of nodes by an integral continued C-fraction is studied. The necessary and sufficient conditions for its solvability are found. As a particular case, the considered integral continued…
The Gibbs--Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions is investigated. We prove existence of the overshoot characteristic of the phenomenon for certain cardinal…
The objective of the paper is to study accuracy of multi-class classification in high-dimensional setting, where the number of classes is also large ("large $L$, large $p$, small $n$" model). While this problem arises in many practical…
The simplest way to obtain continuous interpolation between two points in high dimensional space is to draw a line between them. While previous works focused on the general connectivity between model parameters, we explored linear…
The aim of this expository article is to present recent developments in the centuries old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include,…
We show for $A,B\subset\mathbb{R}^d$ of equal volume and $t\in (0,1/2]$ that if $|tA+(1-t)B|< (1+t^d)|A|$, then (up to translation) $|\text{co}(A\cup B)|/|A|$ is bounded. This establishes the sharp threshold for Figalli and Jerison's…