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The best rank-R1 R2 R3 approximation of tensors by means of a geometric Newton method

Summary: The best rank-´R1 R2 R3µ approximation of tensors by means
of a geometric Newton method
Mariya Ishteva£
, Lieven De Lathauwer£,
, P.-A. Absil££
and Sabine Van Huffel£
ESAT/SCD, Katholieke Universiteit Leuven, Belgium

Subfaculty Science and Technology, Katholieke Universiteit Leuven Campus Kortrijk, Belgium
INMA, Université catholique de Louvain, Belgium
Abstract. In the matrix case, the best low-rank approximation can be obtained directly from the truncated singular value
decomposition. However, the best rank-´R1 R2 RNµ approximation of higher-order tensors cannot be computed in a
straightforward way. An additional difficulty comes from an invariance property of the cost function by the action of the
orthogonal group. This means that the zeros of the cost function are not isolated and the plain Newton method might have
difficulties. In this paper, we derive a geometric Newton method using the theory of quotient manifolds. Our algorithm has
fast convergence in a neighborhood of the solution and is useful for a large class of problems. We mention some applications.
Keywords: higher-order tensor, rank-´R1 R2 RN µ reduction, quotient manifold, differential-geometric optimization
PACS: 02.10.Xm, 02.40.Sf, 02.40.Vh


Source: Absil, Pierre-Antoine - Département d'ingénierie Mathématique, Université Catholique de Louvain
Leuven, Katholieke Universiteit, Department of Electrical Engineering, SCD Division


Collections: Engineering; Mathematics