Pencil-based algorithms for tensor rank decomposition are not stable

I will discuss the existence of an open set of n1× n2× n3 tensors of rank r on which a popular and efficient class of algorithms for computing tensor rank decompositions is numerically unstable. Algorithm of this class are based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition. The analysis shows that the unstability is caused by the fact that the condition number of the tensor rank decomposition can be much larger for n1×n2×2 tensors than for the n1×n2×n3 input tensor. Joint work with Carlos Beltran and Nick Vannieuwenhoven.