Related papers: Inexact subspace projection methods for low-rank t…
New algorithms are proposed for the Tucker approximation of a 3-tensor, that access it using only the tensor-by-vector-by-vector multiplication subroutine. In the matrix case, Krylov methods are methods of choice to approximate the dominant…
Many optimization problems require hyperparameters, i.e., parameters that must be pre-specified in advance, such as regularization parameters and parametric regularizers in variational regularization methods for inverse problems, and…
Tensor completion can estimate missing values of a high-order data from its partially observed entries. Recent works show that low rank tensor ring approximation is one of the most powerful tools to solve tensor completion problem. However,…
Inverse problems arise in various scientific and engineering applications, necessitating robust numerical methods for their solution. In this work, we consider the effectiveness of Krylov subspace iterative methods, including GMRES, QMR,…
This paper is concerned with the regularization of large-scale discrete inverse problems by means of inexact Krylov methods. Specifically, we derive two new inexact Krylov methods that can be efficiently applied to unregularized or…
We propose an isogeometric solver for Poisson problems that combines i)low-rank tensor techniques to approximate the unknown solution and the system matrix, as a sum of a few terms having Kronecker product structure, ii) a Truncated…
Due to the explosive growth of large-scale data sets, tensors have been a vital tool to analyze and process high-dimensional data. Different from the matrix case, tensor decomposition has been defined in various formats, which can be…
We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank…
In this paper, we take a step towards developing efficient hard thresholding methods for low-rank tensor recovery from memory-efficient linear measurements with tensorial structure. Theoretical guarantees for many standard iterative…
We provide convergence rates for Krylov subspace solutions to the trust-region and cubic-regularized (nonconvex) quadratic problems. Such solutions may be efficiently computed by the Lanczos method and have long been used in practice. We…
In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model…
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic…
Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop…
The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and…
Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov's Accelerated Riemannian Gradient algorithm by efficient orthographic retraction…
The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational…
In this paper, we study first-order methods on a large variety of low-rank matrix optimization problems, whose solutions only live in a low dimensional eigenspace. Traditional first-order methods depend on the eigenvalue decomposition at…
Krylov subspace methods are a powerful family of iterative solvers for linear systems of equations, which are commonly used for inverse problems due to their intrinsic regularization properties. Moreover, these methods are naturally suited…
Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such…
Iterative Krylov projection methods have become widely used for solving large-scale linear inverse problems. However, methods based on orthogonality include the computation of inner-products, which become costly when the number of…