Shenglong Hu’s research while affiliated with National University of Defense Technology and other places

In this paper, we present a method to certify the approximation quality of a low rank tensor to a given third order symmetric tensor. Under mild assumptions, best low rank approximation is attained if a control parameter is zero or quantified quasi-optimal low rank approximation is obtained if the control parameter is positive. This is based on a primal-dual method for computing a low rank approximation for a given tensor. The certification is derived from the global optimality of the primal and dual problems, and is characterized by easily checkable relations between the primal and the dual solutions together with another rank condition. The theory is verified theoretically for orthogonally decomposable tensors as well as numerically through examples in the general case.

The quartic minimization over the sphere can be reformulated as a nonlinear nonconvex semidefinite program over the spectraplex. In this paper, under mild assumptions, we show that the reformulated nonlinear semidefinite program possesses the quadratic growth property at a rank one critical point which is a local minimizer of the quartic minimization problem. The quadratic growth property further implies the strong metric subregularity of the subdifferential of the objective function of the unconstrained reformulation of the nonlinear semidefinite program, from which we can show that the objective function is a Łojasiewicz function with exponent 12$\frac{1}{2}$ at the corresponding critical point. With these results, we can establish the linear convergence of an efficient DCA method proposed for solving the nonlinear semidefinite program.

In this paper, we present a method to certify the approximation quality of a low rank tensor to a given third order symmetric tensor. Under mild assumptions, best low rank approximation is attained if a control parameter is zero or quantified quasi-optimal low rank approximation is obtained if the control parameter is positive.This is based on a primal-dual method for computing a low rank approximation for a given tensor. The certification is derived from the global optimality of the primal and dual problems, and is characterized by easily checkable relations between the primal and the dual solutions together with another rank condition. The theory is verified theoretically for orthogonally decomposable tensors as well as numerically through examples in the general case.

In this paper, a method for quartic minimization over the sphere is studied. It is based on an equivalent difference of convex (DC) reformulation of this problem in the matrix variable. This derivation also induces a global optimality certification for the quartic minimization over the sphere. An algorithm with the subproblem being solved by a semismooth Newton method is then proposed for solving the quartic minimization problem. The efficiency of this algorithm and the global optimality certification are illustrated by numerical experiments.

In this paper, a method combining several recent advances is proposed for solving the tensor completion problem. In the proposed method, the ɛ-sparsity index of a matrix, the balanced matricization schemes, the ket augmentation technique, and the discrete cosine transformation are carefully integrated. An easily implementable algorithm is presented to solve the proposed method, and each subproblem can be solved either exactly or approximately by closed formula. The global convergence of the algorithm is established without any assumptions. Experiments with real data show the ability of the proposed method over several state-of-the-art methods, especially with low sample ratios.

Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this paper, an improved version iAPD of the classical APD is proposed. For the first time, all of the following four fundamental properties are established for iAPD: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in optimization; (iii) more importantly, it converges R-linearly for a generic tensor without any assumption; (iv) for almost all LROTA problems, iAPD reduces to APD after finitely many iterations if it converges to a local minimizer.

In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors of tensors is studied, which have found many applications in diverse areas. The main results are: (i) each (Z-)eigenvector/singular vector tuple of a generic tensor is nondegenerate, and (ii) each nonzero Z-eigenvector/singular vector tuple of an orthogonally decomposable tensor is nondegenerate.

In this paper, a complete characterization of the B${\text {B}}$-subdifferential with explicit formula for the projection mapping onto the generalized spectraplex (aka generalized matrix simplex) is derived. The derivation is based on complete characterizations of the B${\text {B}}$-subdifferential as well as the Han-Sun Jacobian of the projection mapping onto the generalized simplex. The formula provides tools for further computations and nonsmooth analysis involving this projection.

Due to the multi-linearity of tensors, most algorithms for tensor optimization problems are designed based on the block coordinate descent method. Such algorithms are widely employed by practitioners for their implementability and effectiveness. However, these algorithms usually suffer from the lack of theoretical guarantee of global convergence and analysis of convergence rate. In this paper, we propose a block coordinate descent type algorithm for the low rank partially orthogonal tensor approximation problem and analyse its convergence behaviour. To achieve this, we carefully investigate the variety of low rank partially orthogonal tensors and its geometric properties related to the parameter space, which enable us to locate KKT points of the concerned optimization problem. With the aid of these geometric properties, we prove without any assumption that: (1) Our algorithm converges globally to a KKT point; (2) For any given tensor, the algorithm exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in nonconvex optimization; {(3)} For a generic tensor, our algorithm converges R-linearly.

An important tool in matrix optimization problems is the strong semismoothness of the projection mapping onto the cone of real symmetric positive semidefinite matrices, and the explicit formula for its ${\text {B}}$(ouligand)-subdifferentials. In this paper, we examine the corresponding results for the so-called matrix simplex, that is, the set of real symmetric positive semidefinite matrices whose traces are equal to one. This result complements the current literature and enlarges the toolbox of matrix spectral operators whose ${\text {B}}$-subdifferentials are explicitly formulated. Since the matrix simplex frequently arises in subproblems for solving matrix optimization problems, the derived results can potentially serve as a useful tool for efficiently solving these problems. As an illustration, we present a numerical example to demonstrate that the proposed approach can outperform the existing approaches which used projection mapping onto positive semidefinite matrix cone directly.