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In this paper we introduce a generic model for multiplicative algorithms which is suitable for the MapReduce parallel programming paradigm. We implement three typical machine learning algorithms to demonstrate how similarity comparison,…

There has been a large increase in the amount of work on hierarchical low-rank approximation methods, where the interest is shared by multiple communities that previously did not intersect. This objective of this article is two-fold; to…

We consider the problem of estimation of a low-rank matrix from a limited number of noisy rank-one projections. In particular, we propose two fast, non-convex \emph{proper} algorithms for matrix recovery and support them with rigorous…

In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…

We study approximation of functions by algebraic polynomials in the H\"older spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of…

In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange…

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…

New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…

Inspired by the latest developments in multilevel Monte Carlo (MLMC) methods and randomised sketching for linear algebra problems we propose a MLMC estimator for real-time processing of matrix structured random data. Our algorithm is…

We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…

Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…

Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time…

Machine learning can provide deep insights into data, allowing machines to make high-quality predictions and having been widely used in real-world applications, such as text mining, visual classification, and recommender systems. However,…

In this paper the main results in arXiv:0901.3179v3, related to the matrix representation of polynomial maps, are restated in traditional way of linear algebra assuming that variable vectors are presented as column vectors. Some new results…

This paper provides the first meaningful documentation and analysis of an established technique which aims to obtain an approximate solution to linear programming problems prior to applying the primal simplex method. The underlying…

Polynomial functions have plenty of useful analytical properties, but they are rarely used as learning models because their function class is considered to be restricted. This work shows that when trained properly polynomial functions can…

In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…

In this paper we propose a method for the approximation of high-dimensional functions over finite intervals with respect to complete orthonormal systems of polynomials. An important tool for this is the multivariate classical analysis of…

Square matrices appear in many machine learning problems and models. Optimization over a large square matrix is expensive in memory and in time. Therefore an economic approximation is needed. Conventional approximation approaches factorize…

Large Language Models have received significant attention due to their abilities to solve a wide range of complex tasks. However these models memorize a significant proportion of their training data, posing a serious threat when disclosed…