相关论文: Chebyshev approximation and composition of functio…

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank…

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents…

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…

This work is concerned with approximating a trivariate function defined on a tensor-product domain via function evaluations. Combining tensorized Chebyshev interpolation with a Tucker decomposition of low multilinear rank yields function…

Tensor trains (TTs), also known as matrix product states (MPS), are compressed representations of high-dimensional data that can be efficiently manipulated to perform calculations on the data. In many applications, such as TT-based solvers…

Quantized tensor trains (QTTs) have recently emerged as a framework for the numerical discretization of continuous functions, with the potential for widespread applications in numerical analysis. However, the theory of QTT approximation is…

Encoding classical data in a quantum state is a key prerequisite of many quantum algorithms. Recently matrix product state (MPS) methods emerged as the most promising approach for constructing shallow quantum circuits approximating input…

Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in mathematics, has been proposed originally for describing an (especially one-dimensional) quantum system, and recently has found applications in various…

Using the matrix product state (MPS) representation of tensor train decompositions, in this paper we propose a tensor completion algorithm which alternates over the matrices (tensors) in the MPS representation. This development is motivated…

We introduce compositional tensor trains (CTTs) for the approximation of multivariate functions, a class of models obtained by composing low-rank functions in the tensor-train format. This format can encode standard approximation tools,…

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…

This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism…

Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These…

Recent work by Wu {\em et al.} [arXiv:1910.11011] proposed a numerical method, so-called matrix product operator-matrix product state (MPO-MPS) method, by which several types of quantum many-body wave functions, in particular, the projected…

Tensor networks are a compressed format for multi-dimensional data. One-dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum…

We show that recursively generated Chebyshev expansions offer numerically efficient representations for calculating zero-temperature spectral functions of one-dimensional lattice models using matrix product state (MPS) methods. The main…

The Chebyshev expansion offers a numerically efficient and easy-implement algorithm for evaluating dynamic correlation functions using matrix product states (MPS). In this approach, each recursively generated Chebyshev vector is…

Multivariate functions of continuous variables arise in countless branches of science. Numerical computations with such functions typically involve a compromise between two contrary desiderata: accurate resolution of the functional…

Tensor networks, which are originally developed for characterizing complex quantum many-body systems, have recently emerged as a powerful framework for capturing high-dimensional probability distributions with strong physical…

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in…