Finding entries of maximum absolute value in low-rank tensors

Abstract: We present an iterative method for the search of extreme entries in low-rank tensors which is based on a power iteration combined with a binary search. In this work we use the HT-format for low-rank tensors but other low-rank formats can be used verbatim. We present two different approaches to accelerate the basic power iteration: an orthogonal projection of Rayleigh-Ritz type, as well as an acceleration of the power iteration itself which can be achieved due to the diagonal structure of the underlying eigenvalue problem. Finally the maximizing index is determined by a binary search based on the proposed iterative method for the approximation of the maximum norm. The iterative method for the maximum norm estimation inherits the linear complexity of the involved tensor arithmetic in the HT-format w.r.t. the tensor order, which is also verified by numerical tests.