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 ∈ RI1×I2...×IN from a subset of its entries which are deter- mined by a selected subset of Rn indices in each mode (n = 1, 2, ..., N ). As a generalization of the column-row ma- trix decomposition (also known as CUR or “skeleton” decom- position), 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 ap- plied 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 sub- tensor determined by the selected indices. We also discuss about its applications for signal processing where low dimen- sional 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.