Related papers: Multigrid methods for tensor structured Markov cha…

Numerical methods based on tensor networks have been extensively explored in the research of quantum many-body systems in recent years. It has been recognized that the ability of tensor networks to describe a quantum many-body state…

We introduce and analyze a mesh-free two-level hybrid Chebyshev-Tucker tensor representation for approximating multivariate functions, which combines tensor-product Chebyshev interpolation with the low-rank Tucker decomposition of the…

The widespread use of multisensor technology and the emergence of big datasets have created the need to develop tools to reduce, approximate, and classify large and multimodal data such as higher-order tensors. While early approaches…

Scientific problems require resolving multi-scale phenomena across different resolutions and learning solution operators in infinite-dimensional function spaces. Neural operators provide a powerful framework for this, using…

This paper is concerned with the approximation of high-dimensional functions in a statistical learning setting, by empirical risk minimization over model classes of functions in tree-based tensor format. These are particular classes of…

We present a Markov-chain analysis of blockwise-stochastic algorithms for solving partially block-separable optimization problems. Our main contributions to the extensive literature on these methods are statements about the Markov operators…

Markov chains are a natural and well understood tool for describing one-dimensional patterns in time or space. We show how to infer $k$-th order Markov chains, for arbitrary $k$, from finite data by applying Bayesian methods to both…

We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically…

This paper introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second order partial differential equations. Based on a discrete…

The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of…

This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of…

This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain $D$ such that…

In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to…

The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that…

The interest in machine learning with tensor networks has been growing rapidly in recent years. We show that tensor-based methods developed for learning the governing equations of dynamical systems from data can, in the same way, be used…

This paper deals with the state estimation of stochastic systems and examines the possible employment of tensor decompositions in grid-based filtering routines, in particular, the tensor-train decomposition. The aim is to show that these…

The use of higher-order stochastic processes such as nonlinear Markov chains or vertex-reinforced random walks is significantly growing in recent years as they are much better at modeling high dimensional data and nonlinear dynamics in…

This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary…

Tensor factorization is a powerful tool to analyse multi-way data. Compared with traditional multi-linear methods, nonlinear tensor factorization models are capable of capturing more complex relationships in the data. However, they are…

Graphical models are popular statistical tools which are used to represent dependent or causal complex systems. Statistically equivalent causal or directed graphical models are said to belong to a Markov equivalent class. It is of great…