Abstract:

Tensor (network) decompositions are a way to parametrize multipartite tensors efficiently, finding applications in quantum many-body physics, machine learning, and probability theory. However, challenges arise when dealing with elements that satisfy positivity constraints, such as density matrices or probability distributions. These challenges include separations between ranks or the undecidability of the description validity.
In this talk, we focus on positive tensor decompositions in the context of approximations. Initially, we show that fixed approximation errors eliminate certain separations between positive ranks. Additionally, we prove that there are instabilities in positive ranks for tensor networks containing a loop, a known issue for unconstrained tensor decompositions. If time permits, we establish a connection between the positivity problem for Matrix Product Operators and a modified version of Skolem's problem. This open problem involves determining whether there exists an algorithm that can decide if a linear recurrence sequence attains specific values.