Structured Tensors and the Geometry of Data

Tensors are higher dimensional analogues of matrices; they are used to record data with multiple changing variables. Interpreting tensor data requires finding low rank structure, and the structure depends on the application or context. In this talk, we describe four projects in the study of structured tensors. Often tensors of interest define semi-algebraic sets, given by polynomial equations and inequalities. We give a characterization of the set of tensors of real rank two, and answer questions about statistical models using probability tensors and semi-algebraic statistics. We also study cubic surfaces as symmetric tensors, and describe current work on learning a path from its three dimensional signature tensor. This talk is based on joint work with Guido Montúfar and Bernd Sturmfels.