This article is cited in 11 papers

The structure of the Hessian and the efficient implementation of Newton's method in the problem of the canonical approximation of tensors

V. A. Kazeev, E. E. Tyrtyshnikov

Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moskow, 119333 Russia

Abstract: A tensor given by its canonical decomposition is approximated by another tensor (again, in the canonical decomposition) of fixed lower rank. For this problem, the structure of the Hessian matrix of the objective function is analyzed. It is shown that all the auxiliary matrices needed for constructing the quadratic model can be calculated so that the computational effort is a quadratic function of the tensor dimensionality (rather than a cubic function as in earlier publications). An economical version of the trust region Newton method is proposed in which the structure of the Hessian matrix is efficiently used for multiplying this matrix by vectors and for scaling the trust region. At each step, the subproblem of minimizing the quadratic model in the trust region is solved using the preconditioned conjugate gradient method, which is terminated if a negative curvature direction is detected for the Hessian matrix.

Key words: tensor decompositions, canonical decomposition, low-rank approximations, trust region Newton method, conjugate gradient method.

UDC: 519.61

Received: 17.12.2009

 English version: Computational Mathematics and Mathematical Physics, 2010, 50:6, 927–945

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