Approximation of High-Order Tensors by Partial Sampling: New Results and Algorithms

CESAR F. CAIAFA; ANDRZEJ CICHOCKI

Pisa, ITALY

Conferencia; ILAS 2010 16 Conference of the International Linear Algebra Society; 2010

International Linear Algebra Society

Recently [1,2,3], a new formula was provided that allows one to reconstruct a rank-(R1 , R2 , ..., RN ) Tucker tensor Y (I1×I2,...,×IN) from a subset of its entries which are determined by a selected subset of Rn indices in each mode (n=1,2,...,N). As a generalization of the column-row matrix decomposition (also known as CUR or skeleton decomposition), which approximates a matrix from a subset of its rows and columns, our result provides a new method for the approximation of a high dimensional (N>=3) tensor by us- ing only the information contained in a subset of its n-mode fibers (n=1,2,...,N). The proposed algorithm can be applied to the case of arbitrary number of dimensions (N>=3) and the indices are sequentially selected in an optimal way based on the previously selected ones. In this talk, we an-alyze and discuss the properties of this method in terms of the subspaces spanned by the unfolding matrices of the subtensor determined by the selected indices. We also discuss about its applications for signal processing where low dimensional signals are mapped to higher dimensional tensors and processed with tensor tools. Experimental results are shown to illustrate the properties and the potential of this method.