The Moduli Space of Generic Tensor Rank Decompositions

This paper explores the intricate structure of the moduli space associated with generic tensor rank decompositions. Tensors, as multidimensional arrays, generalize matrices and find extensive applications across diverse scientific and engineering disciplines. Their decomposition into a sum of rank-one components, particularly the Canonical Polyadic (CP) decomposition, is a fundamental problem. However, unlike matrix rank, tensor rank exhibits complex behaviors, including the lack of uniqueness and the existence of "border rank" phenomena where a tensor is a limit of tensors of lower rank. This work focuses on understanding the moduli space that parameterizes generic tensors and their unique (up to scaling and permutation) minimal rank decompositions. We delve into the algebraic geometric foundations, leveraging concepts such as secant varieties of Segre varieties, and Geometric Invariant Theory (GIT) to characterize the topology and dimension of this moduli space. The properties of this space have profound implications for the identifiability of tensor models, the computational complexity of decomposition algorithms, and the theoretical underpinnings of multilinear algebra. We review existing theoretical results, present a methodology for defining and analyzing such a moduli space, discuss expected results concerning its dimension and singularities, and highlight the practical and theoretical implications for fields reliant on tensor analysis.