Related papers: Generation and application of multivariate polynom…
We study the feature-scaled version of the Monte Carlo algorithm with linear function approximation. This algorithm converges to a scale-invariant solution, which is not unduly affected by states having feature vectors with large norms. The…
We study the problem of minimizing a multivariate polynomial function over the unit hypercube. By representing the polynomial through a hypergraph and exploiting its sparsity structure, we establish a new sufficient condition under which…
A novel algorithm for the computation of the quadratic numerical range is presented and exemplified yielding much better results in less time compared to the random vector sampling method. Furthermore, a bound on the probability for the…
In this contribution we deal with Gaussian quadrature rules based on orthogonal polynomials associated with a weight function $w(x)= x^{\alpha} e^{-x}$ supported on an interval $(0,z)$, $z>0.$ The modified Chebyshev algorithm is used in…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A…
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into…
One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data. A growing body of work studies low-degree polynomials as a restricted model of computation for…
Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses…
In this work, we consider the matrix completion problem, where the objective is to reconstruct a low-rank matrix from a few observed entries. A commonly employed approach involves nuclear norm minimization. For this method to succeed, the…
Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We give an algorithm for this problem that has a running time, and data requirement polynomial in the dimension and…
This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has…
The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a…
We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a…
We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial…
Variational quantum algorithms are poised to have significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these…
In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable…
Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-$ \mathsf{E} $ operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However,…
We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find…
This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high dimensional…