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. Often tensors of interest define semi-algebraic sets, given by polynomial equations and inequalities. I'll 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. I will also describe work on learning a path from its three-dimensional signature tensor. This talk is based on joint work with Guido Montúfar, Max Pfeffer, and Bernd Sturmfels.