Linear algebra is the foundation to methods for finding structure in matrix data. There are many challenges in extending this to the multi-linear setting of tensors. In this talk, I will describe the algebraic properties of tensors that underpin these challenges, as well as progress that has been made. I will focus on extending aspects of the singular value decomposition to tensors: singular vectors, notions of rank, and low-rank approximation. We will also see the analogues of these concepts for symmetric tensors, and connections to classical algebraic geometry.
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