The Best Rank-(r₁, R₂,…, R_N)Approximation Of Higher-Order Tensors

The problem of approximating a given tensor A∈RI1×I2…×IN by another ten-sor B of equal dimensions but of lower rank, occurs in various applications,e.g.,sig-nal processing,analytical chemistry,quantum chemistry, harmonic analysis,independent component (ICA),telecommunications,scientific computing,higher-order statistics,image processing.Similarly to the matrix singular value decomposition (SVD),Lathauwer,Moor and Vandewalle gives the singular value decomposition of the tensor (HOSVD).The best low-rank approximation of the matrix can be obtained from the truncated singular value decomposition (SVD),however,in the tensor case,the truncated higher-order singular val-ue decomposition (HOSVD) gives a suboptimal rank-(r1,r2,…,rN)approximation of the tensor.In this paper, we derive two Newton iterative methods on the Grassmann man-ifold to refine the initial estimates obtained by the truncated higher-order singular value for computing the best rank-(r1,r2,r3) approximation of the3-order tensor.The gener-alization to tensors of order higher than three is straightforward.We compare the Newton algorithm with the known higher-order orthogonal iteration (HOOI) algorithm by numer-ical results.The first Newton method translates the best rank(r1,r2,r3)approximation of the3-order tensor into the zero of the equation of matrix on Grassmann manifold.The sec-ond Newton method gives the best optimal solution by computing the gradient operator and the Hessian operator on the Grassmann manifold.