相关论文: Antithetic variates in higher dimensions
Partial differential equation is a powerful tool to characterize various physics systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper, we present a…
(Mini-batch) Stochastic Gradient Descent is a popular optimization method which has been applied to many machine learning applications. But a rather high variance introduced by the stochastic gradient in each step may slow down the…
This paper concerns the long-standing question of representing (totally) anti-symmetric functions in high dimensions. We propose a new ansatz based on the composition of an odd function with a fixed set of anti-symmetric basis functions. We…
We propose a new algorithm for variance reduction when estimating $f(X_T)$ where $X$ is the solution to some stochastic differential equation and $f$ is a test function. The new estimator is $(f(X^1_T) + f(X^2_T))/2$, where $X^1$ and $X^2$…
Monte Carlo and Quasi-Monte Carlo methods present a convenient approach for approximating the expected value of a random variable. Algorithms exist to adaptively sample the random variable until a user defined absolute error tolerance is…
Monte Carlo simulations are performed for the S = 1/2 XY and ferro- and antiferromagnetic Heisenberg model in two dimensions using the loop algorithm. Thermodynamic properties of all these models are investigated in wide temperature range.…
Orthogonal Monte Carlo (OMC) is a very effective sampling algorithm imposing structural geometric conditions (orthogonality) on samples for variance reduction. Due to its simplicity and superior performance as compared to its Quasi Monte…
Many machine learning problems involve Monte Carlo gradient estimators. As a prominent example, we focus on Monte Carlo variational inference (MCVI) in this paper. The performance of MCVI crucially depends on the variance of its stochastic…
Sparse representations have proven their efficiency in solving a wide class of inverse problems encountered in signal and image processing. Conversely, enforcing the information to be spread uniformly over representation coefficients…
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte…
A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range…
Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix…
Estimating risk measures such as large loss probabilities and Value-at-Risk is fundamental in financial risk management and often relies on computationally intensive nested Monte Carlo methods. While Multi-Level Monte Carlo (MLMC)…
We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new…
Given a positive definite covariance matrix $\widehat \Sigma$, we strive to construct an optimal \emph{approximate} factor analysis model $HH^\top +D$, with $H$ having a prescribed number of columns and $D>0$ diagonal. The optimality…
Let $\mathbb{F}_{q}$ be a finite field of characteristic $2$ and $O_2^+(\mathbb{F}_{q})$ be the $2$-dimensional orthogonal group of plus type over $\mathbb{F}_{q}$. Consider the standard representation $V$ of $O_2^+(\mathbb{F}_{q})$ and the…
The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem.…
An antinorm is a concave analogue of a norm. In contrast to norms, antinorms are not defined on the entire space $R^d$ but on a cone $K\subset R^d$. They are applied in the matrix analysis, optimal control, and dynamical systems. Their…
We generalize Angenent's shrinking tori \cite{Angenent1992} to minimal $n$-dimensional tori embedded in $\mathbb{R}^{n+1}$ equipped with the metric $$g=e^{-\frac{f(\sum^{n+1}_{i=1}x_{i}^{2})}{2n}}\sum^{n+1}_{i=1}dx^{2}_{i},$$ where $f$ is a…
In this paper we present a method for exact generation of multivariate samples with pre-specified marginal distributions and a given correlation matrix, based on a mixture of Fr\'echet-Hoeffding bounds and marginal products. The bivariate…