Related papers: Super-polynomial accuracy of multidimensional rand…
Let $f$ be analytic on $[0,1]$ with $|f^{(k)}(1/2)|\leq A\alpha^kk!$ for some constant $A$ and $\alpha<2$. We show that the median estimate of $\mu=\int_0^1f(x)\,\mathrm{d}x$ under random linear scrambling with $n=2^m$ points converges at…
We study the sample median of independently generated quasi-Monte Carlo estimators based on randomized digital nets and prove it approximates the target integral value at almost the optimal convergence rate for various function spaces. In…
We study a random sampling technique to approximate integrals $\int_{[0,1]^s}f(\mathbf{x})\,\mathrm{d}\mathbf{x}$ by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which…
We consider the problem of evaluating $I(\varphi):=\int_{[0,1)^s}\varphi(x) dx$ for a function $\varphi \in L^2[0,1)^{s}$. In situations where $I(\varphi)$ can be approximated by an estimate of the form $N^{-1}\sum_{n=0}^{N-1}\varphi(x^n)$,…
We study the convergence rate of the median estimator for affine matrix scrambled digital nets applied to integrands over the unit hypercube $[0, 1]^s$. By taking the median of $(2r-1)$ independent randomized quasi-Monte Carlo (RQMC)…
In a very recent work, Basu and Owen (2015) propose the use of scrambled geometric nets in numerical integration when the domain is a product of $s$ arbitrary spaces of dimension $d$ having a certain partitioning constraint. It was shown…
This study investigates the performance of median-of-means sampling compared to traditional mean-of-means sampling for computing the Keister function integral using Randomized Quasi-Monte Carlo (RQMC) methods. The research tests both…
Let us assume that $f$ is a continuous function defined on the unit ball of $\mathbb R^d$, of the form $f(x) = g (A x)$, where $A$ is a $k \times d$ matrix and $g$ is a function of $k$ variables for $k \ll d$. We are given a budget $m \in…
This paper studies the problem of finding the median of N distinct numbers distributed across networked agents. Each agent updates its estimate for the median from noisy local observations of one of the N numbers and information from…
Recent advances in quasi-Monte Carlo integration demonstrate that the median of linearly scrambled digital net estimators achieves near-optimal convergence rates for high-dimensional integrals without requiring a priori knowledge of the…
We study the task of high-dimensional entangled mean estimation in the subset-of-signals model. Specifically, given $N$ independent random points $x_1,\ldots,x_N$ in $\mathbb{R}^D$ and a parameter $\alpha \in (0, 1)$ such that each $x_i$ is…
We investigate $(2+1)$-dimensional discretized directed polymers in Gaussian random media. By numerically calculating the probability distribution function of overlap between two independent and identical systems on a common random…
The {\em Total Influence} ({\em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function \ifnum\plusminus=1 $f: \{\pm1\}^n…
We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating…
This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that…
Let $f:\{-1,1\}^n$ be a polynomial with at most $s$ non-zero real coefficients. We give an algorithm for exactly reconstructing f given random examples from the uniform distribution on $\{-1,1\}^n$ that runs in time polynomial in $n$ and…
We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. {Our…
Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected value and the variance of $\sum_{i=1}^{\lfloor n/2\rfloor}\,d({\boldsymbol{\pi}}(2i-1),{\boldsymbol{\pi}}(2i))$ for a uniformly random permutation ${\boldsymbol{\pi}}$ of…
We study randomized quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a…
In the Random Subset Sum Problem, given $n$ i.i.d. random variables $X_1, ..., X_n$, we wish to approximate any point $z \in [-1,1]$ as the sum of a suitable subset $X_{i_1(z)}, ..., X_{i_s(z)}$ of them, up to error $\varepsilon$. Despite…