Related papers: On Generalizations of the Newton-Raphson-Simpson M…
Given an integer m>=1, let || || be a norm in R^{m+1} and let S denote the set of points with nonnegative coordinates in the unit sphere with respect to this norm. Consider for each 1<= j<= m a function f_j(z) that is analytic in an open…
We investigate in this paper an alternative method to simulation based recursive importance sampling procedure to estimate the optimal change of measure for Monte Carlo simulations. We propose an algorithm which combines (vector and…
The Sum-of-Squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space,…
Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this…
In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the…
Composite function minimization captures a wide spectrum of applications in both computer vision and machine learning. It includes bound constrained optimization, $\ell_1$ norm regularized optimization, and $\ell_0$ norm regularized…
For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details…
The aim of this paper is to establish various factorization results and then to derive estimates for linear functionals through the use of a generalized Taylor theorem. Additionally, several error bounds are established including…
We consider the problem of regularized Poisson Non-negative Matrix Factorization (NMF) problem, encompassing various regularization terms such as Lipschitz and relatively smooth functions, alongside linear constraints. This problem holds…
We investigate the problem of sequential linear data prediction for real life big data applications. The second order algorithms, i.e., Newton-Raphson Methods, asymptotically achieve the performance of the "best" possible linear data…
Bayesian nonparametric mixture models provide a flexible framework for data analysis but are often hindered by the computational expense of traditional inference methods like MCMC. A fast, recursive algorithm proposed by Newton (2002)…
This study proposes a Newton based multiple objective optimization algorithm for hyperparameter search. The first order differential (gradient) is calculated using finite difference method and a gradient matrix with vectorization is formed…
We present a novel game-theoretic formulation of Non-Negative Matrix Factorization (NNMF), a popular data-analysis method with many scientific and engineering applications. The game-theoretic formulation is shown to have favorable scaling…
The behaviour of the generalised Riesz function defined by \[S_{m,p}(x)=\sum_{k=0}^\infty \frac{(-)^{k-1}x^k}{k! \zeta(mk+p)}\qquad (m\geq 1,\ p\geq 1)\] is considered for large positive values of $x$. A numerical scheme is given to compute…
We present an iterative technique for finding zeroes of vector fields on Riemannian manifolds. As a special case we obtain a ``nonlinear averaging algorithm'' that computes the centroid of a mass distribution supported in a set of small…
The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals…
Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have…
We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results…
Non-negative matrix factorization (NMF) is a recently developed technique for finding parts-based, linear representations of non-negative data. Although it has successfully been applied in several applications, it does not always result in…
The Fast Reciprocal Square Root Algorithm is a well-established approximation technique consisting of two stages: first, a coarse approximation is obtained by manipulating the bit pattern of the floating point argument using integer…