Black Box Low Tensor-Rank Approximation Using Fiber-Crosses

Abstract

In this article we introduce a black box type algorithm for the approximation of tensors A in high dimension d. The algorithm adaptively determines the positions of entries of the tensor that have to be computed or read, and using these (few) entries it constructs a low rank tensor approximation X that minimizes the ℓ 2-distance between A and X at the chosen positions. The full tensor A is not required, only the evaluation of A at a few positions. The minimization problem is solved by Newton’s method, which requires the computation and evaluation of the Hessian. For efficiency reasons the positions are located on fiber-crosses of the tensor so that the Hessian can be assembled and evaluated in a data-sparse form requiring a complexity of $\mathcal{O}(Pd)$ , where P is the number of fiber-crosses and d the order of the tensor.