Geometry of rank-one tensors and typical (sub)ranks of order-three tensors

As generalizations of matrices to higher dimensions, tensors are ubiquitous in a wide range of fields such as data science, psychometrics, and various topics in mathematics. In particular, the rank of a tensor T is an important property that quantifies in a sense the degree of independence of the data stored in the d-dimensional table of numbers represented by an order-d tensor. The study of principal component analysis is a powerful tool in data analysis used to estimate the most important trends in a given dataset; mathematically, this is performed by approximating the data by a low-rank matrix. The study of geometric properties of rank-one tensors, known as the Segre variety, and ranks of random order-three tensors is a focus of this thesis. If we think of an order-d tensor as storing the probabilities of different events based on d independent random input variables, the probabilities will be multiplicative, and hence the tensor will have rank one. In this setting, higher rank indicates interdependencies among the variables, which is very common in real-world applications. In addition, data inevitably contains noise, leading to a random component in the entries of the tensor. Even if the variables were to be completely independent, this noise would cause the rank of the tensor to be higher; indeed, the typical ranks of a given tensor size describe the possible ranks that a random tensor could take, and there are even small formats of tensors for which the typical ranks are only known from numerical simulations.