The geometry of the tensor rank decomposition: Perturbation theory

The tensor rank decomposition is an expression of a tensor as a linear combination of rank-1 tensors that appears in a variety of theoretical and practical settings: It finds application in algebraic complexity theory where the length of the rank decomposition of Strassen's tensor corresponds with the multiplicative complexity of matrix multiplication, in algebraic statistics for parameter inference of statistical models, in signal processing for blind source separation, and in chemistry for identifying chemical compounds in an unknown mixture. The tensor rank decomposition is widely employed because of the identifiability property: The expression as a linear combination of rank-1 tensors is generally unique. While identifiability is a requisite property for unambiguously interpreting the unique rank-1 tensors, it does not provide any insight into the geometrical property of stability. For admitting a stable (and, thus, interpretable) rank decomposition, it is additionally required that a small perturbation to the tensor only results in a small perturbation to the individual rank-1 terms. As of present, little is known about this perturbation theory of the tensor rank decomposition. In this project, we will develop such a perturbation theory, founded on the interplay between algebraic geometry, linear algebra, and numerical analysis. Prior research of the applicant on generic identifiability enabled the study of the stronger geometrical property of stability.