相关论文: First-order methods on bounded-rank tensors conver…

In a series of papers \cite{LSJR16, PP17, LPP}, it was established that some of the most commonly used first order methods almost surely (under random initializations) and with step-size being small enough, avoid strict saddle points, as…

We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric…

Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

We present a simple, accurate method for solving consistent, rank-deficient linear systems, with or without addi- tional rank-completing constraints. Such problems arise in a variety of applications, such as the computation of the…

Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a…

In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is…

New versions and extensions of Benson's outer approximation algorithm for solving linear vector optimization problems are presented. Primal and dual variants are provided in which only one scalar linear program has to be solved in each…

In this paper we discuss the notion of singular vector tuples of a complex valued $d$-mode tensor of dimension m_1 x ... x m_d. We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the…

We introduce deterministic perturbation schemes for the recently proposed random directions stochastic approximation (RDSA) [17], and propose new first-order and second-order algorithms. In the latter case, these are the first second-order…

It has been shown that many first-order methods satisfy the perturbed Fenchel duality inequality, which yields a unified derivation of convergence. More first-order methods are discussed in this paper, e.g., dual averaging and bundle…

Gradient-descent based iterative algorithms pervade a variety of problems in estimation, prediction, learning, control, and optimization. Recently iterative algorithms based on higher-order information have been explored in an attempt to…

In this paper, we consider the rank-one tensor completion problem. We address the question of existence and uniqueness of the rank-one solution. In particular we show that the global uniqueness over the field of real numbers can be verified…

We study the basic computational problem of detecting approximate stationary points for continuous piecewise affine (PA) functions. Our contributions span multiple aspects, including complexity, regularity, and algorithms. Specifically, we…

In this paper we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient condition, with or without a quadratic functional growth property. These models include the…

We consider the problem of recovering a low-multilinear-rank tensor from a small amount of linear measurements. We show that the Riemannian gradient algorithm initialized by one step of iterative hard thresholding can reconstruct an…

In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on…

The existing randomized algorithms need an initial estimation of the tubal rank to compute a tensor singular value decomposition. This paper proposes a new randomized fixedprecision algorithm which for a given third-order tensor and a…

In this paper, we propose practical normalized stochastic first-order methods with Polyak momentum, multi-extrapolated momentum, and recursive momentum for solving unconstrained optimization problems. These methods employ dynamically…

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…