Related papers: Polynomial methods in statistical inference: theor…
We consider the problem of efficient integration of an n-variate polynomial with respect to the Gaussian measure in R^n and related problems of complex integration and optimization of a polynomial on the unit sphere. We identify a class of…
We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we…
Probabilistic mixture models have been widely used for different machine learning and pattern recognition tasks such as clustering, dimensionality reduction, and classification. In this paper, we focus on trying to solve the most common…
The Bernstein polynomial basis sees significant use owing to its unique properties, particularly in the field of optimal control. However, the basis is known to have a slow rate of convergence to the function it approximates. With this in…
Orthogonal polynomial approximations form the foundation to a set of well-established methods for uncertainty quantification known as polynomial chaos. These approximations deliver models for emulating physical systems in a variety of…
We describe two algorithms for the eigenvalue, eigenvector problem which, on input a Gaussian matrix with complex entries, finish with probability 1 and in average polynomial time.
These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we…
Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to (i) very restricted model classes where exact or approximate…
The problems of computational data processing involving regression, interpolation, reconstruction and imputation for multidimensional big datasets are becoming more important these days, because of the availability of data and their widely…
Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics,…
We present a coherent Bayesian framework for selection of the most likely model from the five genetic models (genotypic, additive, dominant, co-dominant, and recessive) commonly used in genetic association studies. The approach uses a…
We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, and this work was performed by individuals from many…
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…
Gaussian mixture models are universal approximators in the sense that any smooth density can be approximated arbitrarily well with a Gaussian mixture model with enough components. Due to their broad expressive power, Gaussian mixture models…
Probabilistic graphical models are widely used to model complex systems under uncertainty. Traditionally, Gaussian directed graphical models are applied for analysis of large networks with continuous variables as they can provide…
Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation…
Optimal control under uncertainty is a prevailing challenge for many reasons. One of the critical difficulties lies in producing tractable solutions for the underlying stochastic optimization problem. We show how advanced approximate…
We study efficient algorithms for linear regression and covariance estimation in the absence of Gaussian assumptions on the underlying distributions of samples, making assumptions instead about only finitely-many moments. We focus on how…