Higher-order principal component pursuit via tensor approximation and convex optimization

Recovering the low-rank structure of data matrix from sparse errors arises in the principal component pursuit （PCP）. This paper exploits the higher-order generalization of matrix recovery, named higher-order principal component pursuit （HOPCP）, since it is critical in multi-way data analysis. Unlike the convexification （nuclear norm） for matrix rank function, the tensorial nuclear norm is stil an open problem. While existing preliminary works on the tensor completion field provide a viable way to indicate the low complexity estimate of tensor, therefore, the paper focuses on the low multi-linear rank tensor and adopt its convex relaxation to formulate the convex optimization model of HOPCP. The paper further propose two algorithms for HOPCP based on alternative minimization scheme： the augmented Lagrangian alternating direction method （ALADM） and its truncated higher-order singular value decomposition （ALADM-THOSVD） version. The former can obtain a high accuracy solution while the latter is more efficient to handle the computationally intractable problems. Experimental results on both synthetic data and real magnetic resonance imaging data show the applicability of our algorithms in high-dimensional tensor data processing.

APPROXIMATING THE STATIONARY BELLMAN EQUATION BY HIERARCHICAL TENSOR PRODUCTS

We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law.The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations(PDE),which means that the Bellman equation suffers from the curse of dimensionality.Its non linearity is handled by the Policy Iteration algorithm,where the problem is reduced to a sequence of linear equations,which remain the computational bottleneck due to their high dimensions.We reformulate the linearized Bellman equations via the Koopman operator into an operator equation,that is solved using a minimal residual method.Using the Koopman operator we identify a preconditioner for operator equation,which deems essential in our numerical tests.To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats,in particular tensor trains(TT tensors)and multi-polynomials,together with high-dimensional quadrature,e.g.Monte-Carlo.By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.

Best rank one approximation of real symmetric tensors can be chosen symmetric

We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.

An approximate solution of green's tensor to homogeneous and transversely isotropic media

Fourier transform method is used to obtain an approximate solution of Green's tensor to homogeneous and transversely isotropic media like unidirectional fiber re-inforced composites and austenitic stainless steel materials in order to provide the theoretical basis for the scattering problems. A comparison to homogeneously isotropic media is presented and a brief discussion of the main features of the solution is given