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Yuning Yang

Publications and source records attributed to Yuning Yang.

26 records · Page 2Linked to original sources
arXiv2020

Half-Quadratic Alternating Direction Method of Multipliers for Robust Orthogonal Tensor Approximation

Higher-order tensor canonical polyadic decomposition (CPD) with one or more of the latent factor matrices being columnwisely orthonormal has been well studied in recent years. However, most existing models penalize the noises, if occurring, by employing the least squares loss, which may be sensitive to non-Gaussian noise or outliers, leading to bias estimates of the latent factors. In this paper, based on the maximum a posterior estimation, we derive a robust orthogonal tensor CPD model with Cauchy loss, which is resistant to heavy-tailed noise or outliers. By exploring the half-quadratic property of the model, a new method, which is termed as half-quadratic alternating direction method of multipliers (HQ-ADMM), is proposed to solve the model. Each subproblem involved in HQ-ADMM admits a closed-form solution. Thanks to some nice properties of the Cauchy loss, we show that the whole sequence generated by the algorithm globally converges to a stationary point of the problem under consideration. Numerical experiments on synthetic and real data demonstrate the efficiency and robustness of the proposed model and algorithm.

math.OC↗
arXiv2019

The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence

The epsilon alternating least squares ($ε$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is shown that the algorithm globally converges to a KKT point for all tensors without any assumption. For the original ALS, by further studying the properties of the polar decomposition, we also establish its global convergence under a reality assumption not stronger than those in the literature. These results completely address a question concerning the global convergence raised in [L. Wang, M. T. Chu and B. Yu, \emph{SIAM J. Matrix Anal. Appl.}, 36 (2015), pp. 1--19]. In addition, an initialization procedure is proposed, which possesses a provable lower bound when the number of columnwisely orthonormal factors is one. Armed with this initialization procedure, numerical experiments show that the $ε$-ALS exhibits a promising performance in terms of efficiency and effectiveness.

math.OC↗
arXiv2019

Efficiently Maximizing a Homogeneous Polynomial over Unit Sphere without Convex Relaxation

This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order, which is nonconvex and NP-hard. Recent advances show that semidefinite relaxation is quite effective to find a global solution. However, the solution methods involve full/partial eigenvalue decomposition during the iterates, which heavily limits its efficiency and scalability. On the other hand, for odd degree (odd order) cases, the order has to be increased to even, which potentially reduces the efficiency. To find the global solutions, instead of convexifying the problem, we equivalently reformulate the problem as a nonconvex matrix program based on an equivalence property between symmetric rank-1 tensors and matrices of any order, which is a generalization of the existing results. The program is directly solved by a vanilla alternating direction method, which only involves the computation of leading eigenvalue/singular value of certain matrices, benefiting from the special structure of the program. Although being nonconvex, under certain hypotheses, it is proved that the algorithm converges to a leading eigenvalue of the associated tensor. Numerical experiments on different classes of tensors demonstrate that the proposed approach has a significant improvement in efficiency and scalability, while it can keep the effectiveness of semidefinite relaxation as much as possible. For instance, the proposed method finds the leading eigenpair of a third-order 500 dimensional Hilbert tensor in a personal computer within 100 seconds.

math.OC↗
arXiv2015

Higher order Matching Pursuit for Low Rank Tensor Learning

Low rank tensor learning, such as tensor completion and multilinear multitask learning, has received much attention in recent years. In this paper, we propose higher order matching pursuit for low rank tensor learning problems with a convex or a nonconvex cost function, which is a generalization of the matching pursuit type methods. At each iteration, the main cost of the proposed methods is only to compute a rank-one tensor, which can be done efficiently, making the proposed methods scalable to large scale problems. Moreover, storing the resulting rank-one tensors is of low storage requirement, which can help to break the curse of dimensionality. The linear convergence rate of the proposed methods is established in various circumstances. Along with the main methods, we also provide a method of low computational complexity for approximately computing the rank-one tensors, with provable approximation ratio, which helps to improve the efficiency of the main methods and to analyze the convergence rate. Experimental results on synthetic as well as real datasets verify the efficiency and effectiveness of the proposed methods.

stat.ML↗
arXiv2012

On some properties of nonnegative weakly irreducible tensors

In this paper, we mainly focus on how to generalize some conclusions from nonnegative irreducible tensors to nonnegative weakly irreducible tensors. To do so, a basic and important lemma is proven using new tools. First, we give the definition of stochastic tensors. Then we show that every nonnegative weakly irreducible tensor with spectral radius being one is diagonally similar to a unique weakly irreducible stochastic tensor. Based on it, we prove some important lemmas, which help us to generalize the results related. Some counterexamples are provided to show that some conclusions for nonnegative irreducible tensors do not hold for nonnegative weakly irreducible tensors.

math.SP↗
arXiv2011

An algorithm to find the spectral radius of nonnegative tensors and its convergence analysis

In this paper we propose an iterative algorithm to find out the spectral radius of nonnegative tensors. This algorithm is an extension of the smoothing method for finding the largest eigenvalue of a nonnegative matrix \cite{s14}. For nonnegative irreducible tensors, we establish the converges of the algorithm. Finally we report some numerical results and conclude this paper with some remarks.

math.SP↗