On the decomposition of tensors that are given implicitly

In practice, tensor decomposition algorithms commonly try to minimize the least-squares error between the model and a given tensor. However, in some cases, the tensor is not available as such, but is given only implicitly as a solution of a linear system or is defined by a subspace of a matrix. Such problems appear, for example, in multilinear classification, tensor regression, compressed sensing, blind system identification, systems of polynomial equations and algebraic tensor decomposition algorithms. As the explicit construction of the tensor may not be possible due to nonuniqueness or may be undesirable because of the curse of dimensionality, new types of algebraic and optimization-based algorithms are required. In this talk, I will discuss the challenges encountered when developing algorithms for these implicitly given tensors, and I will illustrate the results for some applications.