Seminari scientifici del dipartimento di Matematica e Informatica location:Universita degli Studi di Perugia, Perugia, Italia date:June 7, 2013
The Schmidt-Eckart-Young theorem for matrices (order two tensors) states that the optimal rank-r approximation to a matrix is obtained by retaining the first r terms of the singular value decomposition, which always exists. In this talk, we consider a generalization of this optimal truncation property for the CANDECOMP/PARAFAC decomposition for tensors of higher order. A necessary orthogonality condition is presented, and, using a dimensionality argument involving the underlying algebraic varieties, it is shown that the Schmidt--Eckart--Young theorem cannot be extended to generic tensors of small rank. We also investigate a probabilistic numerical algorithm to compute the dimension of the aforementioned algebraic variety, revealing a general algorithm to test with high probability the singularity of any structured matrix admitting an efficient matrix-vector product implementation, using only inexact arithmetic.