arXiv subjects

Nicholas F. Marshall

Publications and source records attributed to Nicholas F. Marshall.

25 records · Page 2Linked to original sources

arXiv2019

Randomized mixed Hölder function approximation in higher-dimensions

The purpose of this paper is to extend the result of arXiv:1810.00823 to mixed Hölder functions on $[0,1]^d$ for all $d \ge 1$. In particular, we prove that by sampling an $α$-mixed Hölder function $f : [0,1]^d \rightarrow \mathbb{R}$ at $\sim \frac{1}{\varepsilon} \left(\log \frac{1}{\varepsilon} \right)^d$ independent uniformly random points from $[0,1]^d$, we can construct an approximation $\tilde{f}$ such that $$ \|f - \tilde{f}\|_{L^2} \lesssim \varepsilon^α\left(\log \textstyle{\frac{1}{\varepsilon}} \right)^{d-1/2}, $$ with high probability.

math.CA↗

arXiv2019

Approximating mixed Hölder functions using random samples

Suppose $f : [0,1]^2 \rightarrow \mathbb{R}$ is a $(c,α)$-mixed Hölder function that we sample at $l$ points $X_1,\ldots,X_l$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_1),\ldots,f(X_l)$ be given. If $l \ge c_1 n \log^2 n$, then we can compute an approximation $\tilde{f}$ such that $$ \|f - \tilde{f} \|_{L^2} = \mathcal{O}(n^{-α} \log^{3/2} n), $$ with probability at least $1 - n^{2 -c_1}$, where the implicit constant only depends on the constants $c > 0$ and $c_1 > 0$.

math.CA↗

arXiv2018

The Stability of the First Neumann Laplacian Eigenfunction Under Domain Deformations and Applications

The robustness of manifold learning methods is often predicated on the stability of the Neumann Laplacian eigenfunctions under deformations of the assumed underlying domain. Indeed, many manifold learning methods are based on approximating the Neumann Laplacian eigenfunctions on a manifold that is assumed to underlie data, which is viewed through a source of distortion. In this paper, we study the stability of the first Neumann Laplacian eigenfunction with respect to deformations of a domain by a diffeomorphism. In particular, we are interested in the stability of the first eigenfunction on tall thin domains where, intuitively, the first Neumann Laplacian eigenfunction should only depend on the length along the domain. We prove a rigorous version of this statement and apply it to a machine learning problem in geophysical interpretation.

math.SP↗

arXiv2018

Stretching convex domains to capture many lattice points

We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant $1$. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the $(d-1)$-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes & Freitas, van den Berg, Bucur & Gittins, Ariturk & Laugesen, van den Berg & Gittins, and Gittins & Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =πr^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.

math.MG↗

arXiv2018

Manifold learning with bi-stochastic kernels

In this paper we answer the following question: what is the infinitesimal generator of the diffusion process defined by a kernel that is normalized such that it is bi-stochastic with respect to a specified measure? More precisely, under the assumption that data is sampled from a Riemannian manifold we determine how the resulting infinitesimal generator depends on the potentially nonuniform distribution of the sample points, and the specified measure for the bi-stochastic normalization. In a special case, we demonstrate a connection to the heat kernel. We consider both the case where only a single data set is given, and the case where a data set and a reference set are given. The spectral theory of the constructed operators is studied, and Nyström extension formulas for the gradients of the eigenfunctions are computed. Applications to discrete point sets and manifold learning are discussed.

stat.ML↗

arXiv2018

Triangles capturing many lattice points

We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{>0}^2$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed nontrivial and contains infinitely many elements. We also show that there exist `bad' areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3, 3]$ and has Minkowski dimension at most $3/4$.

math.CO↗

arXiv2017

Time Coupled Diffusion Maps

We consider a collection of $n$ points in $\mathbb{R}^d$ measured at $m$ times, which are encoded in an $n \times d \times m$ data tensor. Our objective is to define a single embedding of the $n$ points into Euclidean space which summarizes the geometry as described by the data tensor. In the case of a fixed data set, diffusion maps (and related graph Laplacian methods) define such an embedding via the eigenfunctions of a diffusion operator constructed on the data. Given a sequence of $m$ measurements of $n$ points, we construct a corresponding sequence of diffusion operators and study their product. Via this product, we introduce the notion of time coupled diffusion distance and time coupled diffusion maps which have natural geometric and probabilistic interpretations. To frame our method in the context of manifold learning, we model evolving data as samples from an underlying manifold with a time dependent metric, and we describe a connection of our method to the heat equation over a manifold with time dependent metric.

math.CA↗