Related papers: Tensor train completion: local recovery guarantees…

We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th…

In this paper, we investigate the statistical convergence rate of a Bayesian low-rank tensor estimator. Our problem setting is the regression problem where a tensor structure underlying the data is estimated. This problem setting occurs in…

Using the matrix product state (MPS) representation of the recently proposed tensor ring decompositions, in this paper we propose a tensor completion algorithm, which is an alternating minimization algorithm that alternates over the factors…

Tensor completion is a core machine learning algorithm used in recommender systems and other domains with missing data. While the matrix case is well-understood, theoretical results for tensor problems are limited, particularly when the…

Tensor completion estimates missing components by exploiting the low-rank structure of multi-way data. The recently proposed methods based on tensor train (TT) and tensor ring (TR) show better performance in image recovery than classical…

We analyze convergence of gradient-descent methods on Riemannian manifolds. In particular, we study randomization of Riemannian gradient algorithms for minimizing smooth cost functions (of Morse-Bott type). We prove that randomized gradient…

This work proposes a novel tensor train random projection (TTRP) method for dimension reduction, where pairwise distances can be approximately preserved. Our TTRP is systematically constructed through a tensor train (TT) representation with…

The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially…

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y$. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the…

Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for…

We develop a new Riemannian descent algorithm that relies on momentum to improve over existing first-order methods for geodesically convex optimization. In contrast, accelerated convergence rates proved in prior work have only been shown to…

One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study $n_1 \times n_2 \times n_3$ third-order tensor completion based on…

Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent…

We study the convergence rate of gradient-based local search methods for solving low-rank matrix recovery problems with general objectives in both symmetric and asymmetric cases, under the assumption of the restricted isometry property.…

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 consider the minimization of a cost function $f$ on a manifold $M$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance $\varepsilon$.…

In low-rank tensor completion tasks, due to the underlying multiple large-scale singular value decomposition (SVD) operations and rank selection problem of the traditional methods, they suffer from high computational cost and high…

Higher-order tensors arise frequently in applications such as neuroimaging, recommendation system, social network analysis, and psychological studies. We consider the problem of low-rank tensor estimation from possibly incomplete,…

Convergence guarantees for optimization over bounded-rank matrices are delicate to obtain because the feasible set is a non-smooth and non-convex algebraic variety. Existing techniques include projected gradient descent, fixed-rank…

We consider Riemannian optimization problems with inequality and equality constraints and analyze a class of Riemannian interior point methods for solving them. The algorithm of interest consists of outer and inner iterations. We show that,…